eric sonnendruc ker max-planck-institut fur plasmaphysik

Numerical methods for the Vlasov equations

Eric Sonnendrucker

Max-Planck-Institut fur Plasmaphysik

Technische Universitat Munchen

Lecture notes

Wintersemester 2012/2013

version of January 14, 2013

Contents

Chapter 1. Introduction 51. Plasmas 52. Controlled thermonuclear fusion 53. The ITER project 74. The Vlasov-Maxwell equations 8

Chapter 2. A hierarchy of models for plasmas 91. The N -body model 92. Kinetic models 103. Fluid models 12

Chapter 3. Some theory on Vlasov systems 171. The linear Vlasov equation 172. The Vlasov-Poisson system 21

Chapter 4. Numerical methods for the Vlasov equation 251. Operator splitting 252. Interpolation 273. Semi-Lagrangian methods 374. Particle methods 47

Bibliography 55

3

CHAPTER 1

Introduction

1. Plasmas

When a gas is brought to a very high temperature (104K or more) electrons leavetheir orbit around the nuclei of the atom to which they are attached. This gives anoverall neutral mixture of charged particles, ions and electrons, which is called plasma.Plasmas are considered beside solids, liquids and gases, as the fourth state of matter.

You can also get what is called a non-neutral plasma, or a beam of charged particles,by imposing a very high potential difference so as to extract either electrons or ionsof a metal chosen well. Such a device is usually located in the injector of a particleaccelerator.

The use of plasmas in everyday life have become common. These include, for exam-ple, neon tubes and plasma displays. There are also a number industrial applications:amplifiers in telecommunication satellites, plasma etching in microelectronics, produc-tion of X-rays.

We should also mention that while it is almost absent in the natural state on Earth,except the Northern Lights at the poles, the plasma is 99% of the mass of the visibleuniverse. Including the stars are formed from plasma and the energy they release fromthe process of fusion of light nuclei such as protons. More information on plasmas andtheir applications can be found on the web site http://www.plasmas.org.

2. Controlled thermonuclear fusion

The evolution of energy needs and the depletion of fossil fuels make it essential todevelop new energy sources. According to the well-known formula E = mc2, we canproduce energy by performing a transformation that removes the mass. There are twomain types of nuclear reactions with this. The fission reaction of generating two lighternuclei from the nucleus of a heavy atom and the fusion reaction that is created fromtwo light atoms a heavier nucleus. Fission is used in existing nuclear power plants.Controlled fusion is still in the research stage.

The fusion reaction is the most accessible to fuse nuclei of deuterium and tritium,which are isotopes of hydrogen, for a helium atom and a neutron high energy will beused to produce the heat necessary to manufacture electricity (see Fig. 2).

The temperatures required for thermonuclear fusion exceed one hundred milliondegrees. At these temperatures the electrons are totally freed from their atoms so thatone obtains a gas of electrons and ions which is a totally ionized plasma. To produceenergy, it is necessary that the amplification factor Q which is the ratio of the powerproduced to the external power supplied is greater than one. Energy balance allows forthe Lawson criterion that connects the amplification factor Q the product nTtE where nis the plasma density, T its temperature and tE energy confinement time in the plasma.

Fusion is the basis of the energy of stars in which a confinement at a sufficient densityis provided by their mass. The research on controlled fusion on Earth is consideringtwo approaches. On the one hand inertial confinement fusion aims at achieving a veryhigh density for a relatively short time by shooting on a capsule of deuterium and

5

6 1. INTRODUCTION

Figure 1. Examples of plasmas at different densities and temperatures

Figure 2. The Deuterium-Tritium fusion reaction

tritium beams with lasers. On the other hand magnetic confinement fusion consists inconfining the plasma with a magnetic field at a lower density but for a longer time. Thelatter approach is pursued in the ITER project whose construction has just started atCadarache in the south-eastern France. The plasma is confined in a toroidal-shapedchamber called a tokamak that for ITER is shown in Figure 3.

There are also experimental facilities (NIF in the USA and LMJ in France) are beingbuilt for experimental validation of the concept of inertial confinement fusion using lasers.

3. THE ITER PROJECT 7

Figure 3. Artist view of the ITER Tokamak

Note that an alternative option to lasers for inertial confinement using heavy ionsbeams is also pursued. See http://hif.lbl.gov/tutorial/tutorial.html for moredetails.

More information on fusion can be found on wikipedia http://en.wikipedia.org/

wiki/Inertial_confinement_fusion for inertial fusion and http://en.wikipedia.

org/wiki/Magnetic_confinement_fusion for magnetic fusion.The current record fusion power produced for a deuterium-tritium reaction is equal

to 16 megawatts, corresponding to an amplification factor Q = 0.64. It was obtained inthe JET tokamak in England. It is well established that to obtain an amplification factormuch greater than one, it is necessary to use a greater machine, hence the need for theconstruction of the ITER tokamak, which will contain a plasma volume five times largerthan that of JET, to demonstrate the feasibility of a power plant based on magneticfusion. The amplification factor provided in ITER should be greater than 10.

3. The ITER project

The ITER project is a partnership between the European Union, Japan, China,South Korea, Russia, the United States and India for which an international agreementwas signed November 21, 2006 in Paris. It aims to demonstrate the scientific and techni-cal feasibility of producing electricity from fusion energy for which there are significantresources of fuel and which has a low impact on the environment.

The construction of the ITER tokamak is under way in Cadarache in the south-eastern France and the operational phase is expected to begin in 2019 and last for twodecades. The main objectives of ITER are firstly to achieve an amplification factorgreater than 10 and so really allow the production of energy, secondly to implement andtest the technologies needed for a fusion power plant and finally to test concepts for theproduction of Tritium from Lithium belt used to absorb the energy of neutrons.

If successful the next step called DEMO will be to build a fusion reactor fusion thatwill actually produce energy before moving on to commercial fusion power plants.

More information is available on the web site http://www.iter.org.

8 1. INTRODUCTION

4. The Vlasov-Maxwell equations

We consider in this lecture more specifically one of the models commonly used todescribe the evolution of a plasma and which is called a kinetic model. It is based onthe Vlasov equation which describes the evolution of charged particles in an electromag-netic field which can either be self-consistent, that is to say, generated by the particlesthemselves, or externally applied, or most often, both. It is written for non-relativisticparticles

∂fs∂t

+ v · ∂fs∂x

+q

m(E + v ×B) · ∂fs

∂v= 0,

where m is the mass particles, q their charge and f ≡ f(x,v, t) represents the particledensity in phase space at point (x,v) and at time t. It has the structure of a transportequation in phase space which includes the three dimensions of physical space and thethree dimensions of velocity space (or momentum in the relativistic case). The self-consistent electromagnetic field can be calculated by coupling with Maxwell’s equationwith sources that are the charge densities and current calculated from the particles:

− 1

c2

∂E

∂t+∇× B = µ0 J,

∂B

∂t+∇× E = 0,

∇ · E =ρ

ε0,

∇ · B = 0,

with

ρ(x, t) = q

∫f(x,v, t) dv, J(x, t) = q

∫f(x,v, t)v dv.

Plasmas, in particular fusion plasmas are extremely complex objects, involving non-linear interactions and a large variety of time and space scales. They are subject tomany instabilities and turbulence phenomena that make their confinement challenging.The road to fusion as an energy source therefore requires a very fine understanding ofplasmas using appropriate models and numerical simulations based on these models.

The numerical solution of the three-dimensional Vlasov-Maxwell system is a majorchallenge if only because of the huge size of the system due to the fact that the Vlasovequation is posed in the 6D phase space and the non linear coupling between Vlasovand Maxwell. The seven variables to consider are the three variables giving the positionin physical space and the three variable velocity over time. For the model to be usedin practice, it will be necessary to use reduced models that can be precise enough withrespect to certain characteristics of the studied system: symmetry, small parameters,etc.. Furthermore the specific properties of the Vlasov equation will require the use ofnumerical methods specifically designed for this kind of equations.

CHAPTER 2

A hierarchy of models for plasmas

1. The N-body model

At the microscopic level, a plasma or a particle beam is composed of a number ofparticles that evolve following the laws of classical or relativistic dynamics. So eachparticle obeys Newton’s law

dγmv

dt=∑

Fext,

where m is the mass of the particle, v its velocity γ = (1− |v|2

c2)−

12 is the Lorentz factor (c

being the speed of light). The right hand side Fext is composed of all the forces appliedto the particle, which in our case reduce to the Lorentz force induced by the externaland self-consistent electromagnetic fields. Other forces as the weight of the particles arein general negligible. Whence we have

dγimvidt

=∑j

q(Ej + vi ×Bj).

On the other hand the velocity of a particle vi is linked to its position xi by

dxidt

= vi.

Thus, if the initial positions and velocities of the particles are known as well as theexternal fields, the evolution of the particles is completely determined by the equations

dxidt

= vi,(1)

dγimvidt

=∑j

q(Ej + v ×Bj),(2)

where the sum contains the electric and magnetic field generated by each of the otherparticles as well as the external fields.

Remark 1. This system is Hamiltonian, which can be seen easily in the non rela-tivistic case without magnetic field. In this case the electric field derives from a scalarpotential: E = −∇φ. The hamiltonian then reads

H =v2i

2+

q

mφ.

And sodxidt

=∂H

∂vi= vi,

dvidt

= −∂H∂xi

= − q

m∇φ =

q

mE

The motion of the particles is also hamiltonian in the general case, but one needs totransform to specific coordinates which are called canonical coordinates to exhibit thehamiltonian structure.

9

10 2. A HIERARCHY OF MODELS FOR PLASMAS

In general a plasma consists of a large number of particles, 1010 and more. Themicroscopic model describing the interactions of particles with each other is not used ina simulation because it would be far too expensive. We must therefore find approximatemodels which, while remaining accurate enough can reach a reasonable computationalcost. There is actually a hierarchy of models describing the evolution of a plasma. Thebase model of the hierarchy and the most accurate model is the N -body model wehave described, then there are intermediate models called kinetic and which are basedon a statistical description of the particle distribution in phase space and finally themacroscopic or fluid models that identify each species of particles of a plasma with afluid characterized by its density, its velocity and energy. Fluid models are becoming agood approximation when the particles are close to thermodynamic equilibrium, to whichthey return in long time do to the effects of collisions and for which the distribution ofparticle velocities is a Gaussian.

2. Kinetic models

In a kinetic model, each particle species s in the plasma is characterized by a dis-tribution function fs(x,v, t) which corresponds to a statistical mean of the repartitionof particles in phase space for a large number of realisations of the considered physicalsystem. The product fs dx dv is the average number of particles of species s, whoseposition and velocity are in the box of volume dx dv centred at (x,v).

The distribution function contains much more information than a fluid descriptionas it includes information on the distributions of particle velocities at each position. Akinetic description of a plasma is essential when the distribution function is far awayfrom the Maxwell-Boltzmann distribution (also called Maxwellian) that corresponds tothe thermodynamic equilibrium of plasma. Otherwise a fluid description is sufficient. Inthe limit where the collective effects are dominant on binary collisions between particles,the kinetic equation that is derived, by methods of statistical physics from the N -bodymodel is the Vlasov equation which reads

(3)∂fs∂t

+ v · ∇xfs +qsms

(E + v ×B) · ∇vfs = 0,

in the non relativistic case. In the relativistic case it becomes

(4)∂fs∂t

+ v(p) · ∇xfs + qs(E + v(p)×B) · ∇pfs = 0.

We denote by ∇xfs, ∇vfs and ∇pfs, the respective gradients of fs with respect to thethree position, velocity and momentum variables. The constants qs and ms denote thecharge and mass of the particle species. The velocity is linked to the momentum by therelation v(p) = p

msγs, where γ is the Lorentz factor which can be expressed from the

momentum by γs =√

1 + |p|2m2sc

2 .

This equation expresses that the distribution function f is conserved along the tra-jectories of the particles which are determined by the mean electric field. We denoteby fs,0(x,v) the initial value of the distribution function. The Vlasov equation, whenit takes into account the self-consistent electromagnetic field generated by the parti-cles, is coupled to the Maxwell equations which enable to computed this self-consistent

2. KINETIC MODELS 11

electromagnetic field from the particle distribution:

− 1

c2

∂E

∂t+∇× B = µ0 J,

∂B

∂t+∇× E = 0,

∇ · E =ρ

ε0,

∇ · B = 0.

The source terms for Maxwell’s equation, the charge density ρ(x, t) and the currentdensity J(x, t) can be expressed from the distribution functions of the different speciesof particles fs(x,v, t) using the relations

ρ(x, t) =∑s

qs

∫fs(x,v, t) dv,

J(x, t) =∑s

qs

∫fs(x,v, t)v dv.

Note that in the relativistic case the distribution function becomes a function of positionand momentum (instead of velocity): fs ≡ fs(x,p, t) and charge and current densitiesverify

ρ(x, t) =∑s

qs

∫fs(x,p, t) dp, J(x, t) =

∑s

qs

∫fs(x,p, t)v(p) dp.

When binary collisions between particles are dominant with respect to mean fieldeffects, the distribution function satisfies the Boltzmann equation

∂f

∂t+ v · ∂f

∂x=∑s

Q(f, fs),

where Q is the non linear Boltzmann operator. This operator is sometimes replacedby simpler models. A sum on the collisions with all the species of particles representedby fs, including the particles of the same species, is considered. In many cases not allthe collisions might be considered. In some intermediate cases, the collision operatorappears on the right-hand side of the full Vlasov equation.

The Boltzmann collision operator for two species of particles (that might be identical,in which case fs = f) writes

Q(f, fs)(v) =1

m

∫R3

∫S2

B(|v − v1|, θ) [f(v′)fs(v′1)− f(v)fs(v1)] dv1 dn,

where v′ and v′1 are the velocities after collision of the particles with velocity v and v1

before collision. The deflection angle θ is the angle between v − v1 and v′ − v′1. Thepost-collision velocities are expressed by

v′ = v − 2µ

m[(v − v1) · n]n, v′1 = v1 +

2µ

ms[(v − v1) · n]n,

with µ = mmsm+ms

and n a unit vector on the sphere S2. These expressions are obtainedby writing that momentum and kinetic energy are conserved during a collision. Thecollision kernel B is given. Its precise form depends on the properties of the gas.

Let us briefly sketch out some basic properties of the Boltzmann collision operator,see the book of Cercignani [12] for details.


Proposition 1. For any continuous function ϕ, we have∫R3

Q(f, f)ϕ(v) dv =1

4m

∫R3

∫R3

∫S2

[ϕ(v) + ϕ(v1)− ϕ(v′)− ϕ(v′1)]B(|v − v1|, θ)

[f(v′)f(v′1)− f(v)f(v1)] dv dv1 dn.

From which it follows that ∫R3

Q(f, f)ϕ(v) dv = 0

if and only if ϕ(v) is a linear combination of 1, vx, vy, vz and |v|2.

The first relation is obtained by writing four equal expressions for∫R3 Q(f, f)ϕ(v) dv

obtained by changes of variables conserving |v − v1| and θ so that B(|v − v1|, θ) is notmodified and then expression the integral as the average of the four expressions.

Proposition 2 (Boltzmann inequality). For f > 0 we have∫R3

Q(f, f) ln f dv ≤ 0.

Proposition 3 (H theorem). For f(t,x,v) > 0 a solution of the Vlasov-Boltzmannequation, we define

H(t) =

∫T3

∫R3

f ln f dx dv ≤ 0.

ThendH

dt≤ 0,

and the inequality is strict if f is not of the form f(v) = exp(a + b · v + cv2) (ie aMaxwellian).

A consequence of the H theorem, is that the solution of the Vlasov-Boltzmann equa-tion relaxes when time goes to infinity to a minimum of H which is a Maxwellian.

3. Fluid models

Due to collisions, the particles relax in long time to a Maxwellian, which is a ther-modynamical equilibrium. When this state is approximately attained particles can bedescribed by a fluid like model.

This fluid model can be derived from the Vlasov equations. The fluid model willstill be coupled to Maxwell’s equation for the determination of the self-consistent elec-tromagnetic field.

We start from the Vlasov-Boltzmann equation:

(5)∂f

∂t+ v · ∂f

∂x+

q

m(E + v ×B) · ∂f

∂v= Q(f, f).

Remark 2. The Boltzmann collision operator Q(f, f) on the right hand side isnecessary to provide the relaxation to thermodynamic equilibrium. However it will haveno direct influence on our derivation, as we will consider only the first three velocitymoments which vanish for the Boltzmann operator.

The macroscopic quantities on which the fluid equations will be established aredefined using the first three velocity moments of the distribution function f(x,v, t)

• The particle density is defined by

n(x, t) =

∫f(x,v, t) dv,

3. FLUID MODELS 13

• The mean velocity u(x, t) verifies

n(x, t)u(x, t) =

∫f(x,v, t)v dv,

• The pressure tensor P(x, t) is defined by

P(x, t) = m

∫f(x,v, t)(v − u(x, t))⊗ (v − u(x, t)) dv.

• The scalar pressure is one third of the trace of the pressure tensor

p(x, t) =m

3

∫f(x,v, t)|v − u(x, t)|2 dv,

• The temperature T (x, t) is related to the pressure and the density by

T (x, t) =p(x, t)

n(x, t).

• The energy flux is a vector defined by

Q(x, t) =m

2

∫f(x,v, t)|v|2v(x, t)) dv.

where we denote by |v| =√v · v and for two vectors a = (a1, a2, a3)T and b =

(b1, b2, b3)T , their tensor product a ⊗ b is the 3 × 3 matrix whose components are(aibj)1≤i,j≤3.

We obtain equations relating these macroscopic quantities by taking the first velocitymoments of the Vlasov equation. In the actual computations we shall make use thatf vanishes at infinity and that the plasma is periodic in space. This takes care of allboundary condition problems.

Let us first notice that as v is a variable independent of x, we have v·∇xf = ∇x·(fv).Moreover, as E(x, t) does not depend on v and that the ith component of

v ×B(x, t) =

v2B3(x, t)− v3B2(x, t)v3B1(x, t)− v1B3(x, t)v1B2(x, t)− v2B1(x, t)

is independent of vi, we also have

(E(x, t) + v ×B(x, t)) · ∇vf = ∇v · (f(E(x, t) + v ×B(x, t))).

Integrating the Vlasov equation (5) with respect to velocity v we obtain

∂

∂t

∫f(x,v, t) dv +∇x ·

∫f(x,v, t)v dv + 0 = 0.

Whence, as n(x, t)u(x, t) =∫f(x,v, t)v dv, we get

(6)∂n

∂t+∇x · (nu) = 0.

Multiplying the Vlasov by mv and integrating with respect to v, we get

m∂

∂t

∫f(x,v, t)v dv +m∇x ·

∫(v ⊗ v)f(x,v, t) dv

− q(E(x, t)

∫f(x,v, t) dv +

∫f(x,v, t)v dv ×B(x, t) = 0.

Moreover, ∫v ⊗ vf(x,v, t) dv =

∫(v − u)⊗ (v − u)f(x,v, t) dv + nu⊗ u.


Whence

(7) m∂

∂t(nu) +m∇ · (nu⊗ u) +∇ · P = qn(E + u×B).

Finally multiplying the Vlasov equation by 12m|v|

2 = 12mv · v and integrating with

respect to v, we obtain

1

2m∂

∂t

∫f(x,v, t)|v|2 dv +

1

2m∇x ·

∫(|v|2v)f(x,v, t) dv

+1

2q

∫|v|2∇v · [(E(x, t) + v ×B(x, t))f(x,v, t)] dv = 0.

An integration by parts then yields∫|v|2∇v · (E(x, t) + v ×B(x, t))f(x,v, t) dv

= −2

∫v · [(E(x, t) + v ×B(x, t))f(x,v, t)] dv.

Then, developing∫f |v − u|2 dv we get∫

f |v − u|2 dv =

∫f |v|2 dv − 2u ·

∫vf dv + |u|2

∫f dv =

∫f |v|2 dv − n|u|2,

whence

(8)∂

∂t(3

2p+

1

2mn|u|2) +∇ · Q = E · (qnu).

We could continue to calculate moments of f , but we see that each new expressionreveals a moment of higher order. So we need additional information to have as manyunknowns as equations to solve these equations. This additional information is called aclosure relation.

In our case, we will use as a closure relation the physical property that at ther-modynamic equilibrium the distribution function approaches a Maxwellian distributionfunction that we will note fM (x,v, t) and that can be expressed as a function of themacroscopic quantities n(x, t), u(x, t) and T (x, t) which are the density, mean velocityand temperature of the charged fluid:

fM (x,v, t) =n(x, t)

(2πT (x, t)/m)3/2e− |v−u(x,t)|2

2T (x,t)/m .

We also introduce a classical quantity in plasma physics which is the thermal velocity ofthe particle species considered

vth =

√T

m.

It is easy to verify that the first three moments of the distribution function fM areconsistent with the definition of the macroscopic quantities n, u and T defined for anarbitrary distribution function. We have indeed performing each time the change ofvariable w = v−u

vth ∫fM (x,v, t) dv = n(x, t),∫fM (x,v, t)v dv = n(x, t)u(x, t),∫

fM (x,v, t)|v − u|2 dv = 3n(x, t)T (x, t)/m.

3. FLUID MODELS 15

On the other hand, replacing f by fM in the definitions of the pressure tensor P andthe energy flux Q, we can express these terms also in function of n, u and T whichenables us to obtain a closed system in these three unknowns as opposed to the case ofan arbitrary distribution function f . Indeed, we first notice that, denoting by wi the ith

component of w, ∫wiwje

− |w|2

2 dw =

∣∣∣∣∣ 0 if i 6= j,∫e−|w|2

2 dw if i = j.

It follows that the pressure tensor associated to the Maxwellian is

P = mn

(2πT/m)3/2

∫e− |v−u|2

2T/m (v − u)⊗ (v − u) dv,

and so, thanks to our previous computation, the off diagonal terms of P vanish, and bythe change of variable w = v−u

vth, we get for the diagonal terms

Pii = mn

(2π)3/2

T

m

∫e−

w2

2 w2i dv = nT.

It follows that P = nT I = pI where I is the 3 × 3 identity matrix. It now remains tocompute in the same way Q as a function of n, u and T for the Maxwellian with thesame change of variables, which yields

Q =m

2

n

(2πT/m)3/2

∫e− |v−u|2

2T/m |v|2v(x, t)) dv,

=m

2

n

(2π)3/2

∫e−

w2

2 (vthw + u)2(vthw + u) dw,

=m

2

n

(2π)3/2

∫e−

w2

2 (v2thw

2u + 2v2thu ·ww + |u|2 u) dw,

=m

2n(3

T

mu + 2

T

mu + |u|2u),

as the odd moments in w vanish. We finally get

Q =5

2nTu +

m

2n|u|2u =

5

2pu +

m

2n|u|2u.

Then, plugging the expressions of P and of Q in (6)-(7)-(8) we obtain the fluid equationsfor one species of particles of a plasma:

∂n

∂t+∇x · (nu) = 0(9)

m∂

∂t(nu) +m∇ · (nu⊗ u) +∇p = qn(E + u×B)(10)

∂

∂t(3

2p+

1

2mn|u|2) +∇ · (

5

2pu +

m

2n|u|2u) = E · (qnu),(11)

which corresponds in three dimensions to a system of 5 scalar equation with 5 scalarunknowns which are the density n, the three components of the mean velocity u and thescalar pressure p. These equations need of course to be coupled to Maxwell’s equationsfor the computation of the self-consistent electromagnetic field with, in the case of onlyone particle species ρ = q n and J = q nu. Let us also point out that an approximationoften used in plasma physics is that of a cold plasma, for which T = 0 and thus p = 0.Only the first two equations are needed in this case.

CHAPTER 3

Some theory on Vlasov systems

1. The linear Vlasov equation

The Vlasov equation is a linear scalar hyperbolic partial differential equation whenE and B are assumed to be known independently of f . Setting all constants to one theVlasov can then be written

(12)∂f

∂t+ v · ∇xf + (E + v ×B) · ∇vf = 0,

where E(x, t) and B(x, t) are given fields. Setting

A(x,v, t) =

(v

E + v ×B

),

equation (12) becomes

(13)∂f

∂t+ A · ∇(x,v)f = 0.

Hence it is a linear advection equation in phase space. Moreover

∇(x,v) ·A = ∇x · v +∇v · (E + v ×B)

= 0 +∂

∂v1(E1 + v2B3 − v3B2) +

∂

∂v2(E2 + v3B1 − v1B3)

+∂

∂v2(E3 + v1B2 − v2B1)

= 0.

The Vlasov equation can be written in a conservative form

(14)∂f

∂t+∇(x,v) · (Af) = 0,

as ∇(x,v) · (Af) = A · ∇(x,v)f + f ∇(x,v) ·A.

Remark 3. These properties do not rely on the fact that E and B are given inde-pendently of f and are also valid in the non linear case.

The Vlasov equation can thus be written as a classical advection equation

(15)∂f

∂t+ A · ∇f = 0,

with f : Rd × R+ → R and A : Rd × R+ → Rd.Consider now for s ∈ R+ given, the differential system

dX

dt= A(X, t),(16)

X(s) = x,(17)

which is naturally associated to the advection equation (15).

17

18 3. SOME THEORY ON VLASOV SYSTEMS

Definition 1. The solutions of the system (16) are called characteristics of thelinear advection equation (15). We denote by X(t; s,x) the solution of (16) - (17).

Let us recall the classical theorem of the theory of ordinary differential equations(ODE) which gives existence and uniqueness of the solution of (16)-(17). The proof canbe found in [3] for example.

Theorem 1. Assume that A ∈ Ck−1(Rd× [0, T ]), ∇A ∈ Ck−1(Rd× [0, T ]) for k ≥ 1and that

|A(x, t)| ≤ κ(1 + |x|) ∀t ∈ [0, T ] ∀x ∈ Rd.

Then for all s ∈ [0, T ] and x ∈ Rd, there exists a unique solution X ∈ Ck([0, T ]t ×[0, T ]s × Rdx) of (16) - (17).

Proposition 4. Under the hypotheses of the previous theorem we have the followingproperties:

(i) ∀t1, t2, t3 ∈ [0, T ] and ∀x ∈ Rd

X(t3; t2,X(t2; t1,x)) = X(t3; t1,x).

(ii) ∀(t, s) ∈ [0, T ]2, the application x 7→ X(t; s,x) is a C1- diffeomorphism of Rdof inverse y 7→ X(s; t,y).

(iii) The jacobian J(t; s, 1) = det(∇xX(t; s,x)) verifies

∂J

∂t= (∇ ·A)(t;X(t; s,x))J,

and J > 0. In particular if ∇ ·A = 0, J(t; s, 1) = J(s; s, 1) = det Id = 1, whereId is the identity matrix of order d.

Proof. (i) The points x = X(t1; t1,x), X(t2; t1,x), X(t3; t1,x) are on thesame characteristic curve. This curve is characterized by the initial conditionX(t1) = x. So, taking any of these points as initial condition at the corre-sponding time, we get the same solution of (16)-(17). We have in particularX(t3; t2,X(t2; t1,x)) = X(t3; t1,x).

(ii) Taking t1 = t3 in the equality (i) we have

X(t3; t2,X(t2; t3,x)) = X(t3; t3,x) = x.

Hence X(t3; t2, .) is the inverse of X(t2; t3, .) (we denote by g(.) the function x 7→g(x)) and both applications are of class C1 because of the previous theorem.

(iii) Let

J(t; s, 1) = det(∇xX(t; s,x)) = det((∂Xi(t; s,x)

∂xj))1≤i,j≤d.

But X verifies dXdt = A(X(t), t). So we get in particular taking the ith line of

this equality dXidt = Ai(X(t), t). And taking the gradient we get

d

dt∇Xi =

d∑k=1

∂Ai∂xk∇Xk.

For a d×d matrix M the determinant of M is a d-linear alternated form takingas arguments the columns of M . So, denoting by (., . . . , .) this alternated d-linear form, we can write detM = (M1, . . . ,Md) where Mj is the jth column of

1. THE LINEAR VLASOV EQUATION 19

M . Using this notation in our case, we get

∂J

∂t(t; s, 1) =

∂

∂tdet(∇xX(t; s,x))

= (∂∇X1

∂t,∇X2, . . . ,∇Xd) + · · ·+ (∇X1,∇X2, . . . ,

∂∇Xd

∂t)

= (d∑

k=1

∂A1

∂xk∇Xk,∇X2, . . . ,∇Xd) + . . .

+ (∇X1,∇X2, . . . ,d∑

k=1

∂Ad∂xk∇Xk)

=∂A1

∂x1J + · · ·+ ∂Ad

∂xdJ,

as (., . . . , .) is alternated and d-linear. Thus we have ∂J∂t (t; s, 1) = (∇ ·A)J . On

the other hand ∇xX(s; s,x) = ∇xx = Id and so J(s; s, 1) = det Id = 1. J is asolution of the differential equation

dJ

dt= (∇ ·A) J, J(s) = 1,

which admits as the unique solution J(t) = e∫ ts ∇·A dt > 0 and in particular, if

∇ ·A = 0, we have J(t; s, 1) = 1 for all t.

After having highlighted the properties of the characteristics, we can now expressthe solution of the linear advection equation (15) using the characteristics.

Theorem 2. Let f0 ∈ C1(Rd) and A a vector field verifying the hypotheses of theprevious theorem. Then there exists a unique solution of the linear advection equation(15) associated to the initial condition f(x, 0) = f0(x). It is given by

(18) f(x, t) = f0(X(0; t,x)),

where X represent the characteristics associated to A.

Proof. The function f given by (18) is C1 as f0 and X are, and X is defineduniquely. Let’s verify that f is a solution of (15) and that it verifies the initial condition.We first have using formula (18)

f(x, 0) = f0(X(0; 0,x)) = f0(x).

Then∂f

∂t(x, t) =

∂X

∂s(0; t,x) · ∇f0(0; t,x),

and

∇xf(x, t) = ∇x(f0(X(0; t,x))

=

d∑k=1

∂f0

∂xk∇xXk(0; t,x)),

= ∇xX(0; t,x)T∇xf0(X(0; t,x)),

in the sense of a matrix vector product with the jacobian matrix

∇xX(0; t,x) = ((∂Xk

∂xl(0; t,x)))1≤k,l≤d.


We then get

(19) (∂f

∂t+ A · ∇xf)(x, t) =

∂X

∂s(0; t,x) · ∇f0(0; t,x)

+ A(x, t) ·(∇xX(0; t,x)T∇xf0(X(0; t,x))

).

Because of the properties of the characteristics we also have that

X(t; s,X(s; r,x)) = X(t; r,x)

and taking the derivative with respect to s, we get

∂X

∂s(t; s,X(s; r,x)) +∇xX(t; s,X(s; r,x))

∂X

∂t(s; r,x) = 0.

But by definition of the characteristics ∂X∂t (s; r,x) = A(X(s; r,x), s) and as this equation

is verified for all values of t, r, s and so in particular for r = s. It becomes in this case

∂X

∂s(t; s,x) +∇xX(t; s,x)A(x, s) = 0.

Plugging this expression into (19) we obtain

(∂f

∂t+ A · ∇xf)(x, t) = −∇xX(0; t,x)A(x, t)) · ∇f0(X(0; t,x))

+ A(x, t) ·(∇xX(0; t,x)T∇xf0(X(0; t,x))

).

But for a matrix M ∈ Md(R) and two vectors u,v ∈ Rd, on a (Mu) · v = uTMTv =u · (MTv). Whence we get

∂f

∂t+ A · ∇xf = 0,

which means that f defined by (18) is solution of (15).The problem being linear, if f1 and f2 are two solutions we have

∂

∂t(f1 − f2) + A · ∇x(f1 − f2) = 0,

and using the characteristics ddt(f1 − f2)(X(t), t) = 0. So if f1 and f2 verify the same

initial condition, they are identical, which gives the uniqueness of the solution which isthus the function given by formula (18).

Examples.

(1) The free streaming equation

∂f

∂t+ v

∂f

∂x= 0.

The characteristics are solution ofdX

dt= V,

dV

dt= 0.

This we have V (t; s, x, v) = v and X(t; s, x, v) = x+ (t− s)v which gives us thesolution

f(x, v, t) = f0(x− vt, v).

(2) Uniform focusing in a particle accelerator (1D model). We then have E(x, t) =−x and the Vlasov writes

∂f

∂t+ v

∂f

∂x− x∂f

∂v= 0.

dX

dt= V,

dV

dt= −X.

2. THE VLASOV-POISSON SYSTEM 21

Whence get X(t; s, x, v) = x cos(t−s)+v sin(t−s) and V (t; s, x, v) = −x sin(t−s) + v cos(t− s) form which we compute the solution

f(x, v, t) = f0(x cos t− v sin t, x sin t+ v cos t).

2. The Vlasov-Poisson system

2.1. The equations. The Poisson equation is obtained from the Maxwell equa-tions, when the electric and magnetic fields are not, or only very little, time dependent.We then get the stationary Maxwell equations

∇× B = J,(20)

∇× E = 0,(21)

∇ · E = ρ,(22)

∇ · B = 0.(23)

In this case the electric and magnetic fields are decoupled, and in many cases, becauseB itself is small, or because its contribution in the Lorentz force v × B is small, weshall only need the electric field, which is given by equations (21) and (22). Equation(21) implies that E derives from a scalar potential E = −∇φ, and then (22) implies thePoisson equation

−∆φ = ρ,

along with adequate boundary conditions.We consider the dimensionless Vlasov-Poisson equation for one species with a neu-

tralizing background

(24)∂f

∂t+ v · ∇xf −E · ∇vf = 0,

(25) −∆φ = 1− ρ, E = −∇φ,with

ρ(x, t) =

∫f(x,v, t) dv.

The domain on which the system is posed is considered periodic in x and the wholespace R3 in velocity.

We first notice that the Vlasov equation (24) can also be written in conservativeform

(26)∂f

∂t+∇x,v · (Ff) = 0,

with F = (v,−E)T such that ∇x,v · F = 0.

2.2. Conservation properties. The Vlasov-Poisson system has a number of con-servation properties that need special attention when developing numerical methods. Inprinciple it is beneficial to retain the exact invariants in numerical methods and whenit is not possible to keep them all as is the case here, they can be use to monitor thevalidity of the simulation by checking that they are approximately conserved with goodaccuracy.

Proposition 5. The Vlasov-Poisson system verifies the following conservation prop-erties:

• Maximum principle

(27) 0 ≤ f(x,v, t) ≤ max(x,v)

(f0(x,v)).


• Conservation of Lp, norms for p integer, 1 ≤ p ≤ ∞

(28)d

dt

(∫(f(x,v, t))p dx dv

)= 0

• Conservation of volume. For any volume V of phase space

(29)

∫Vf(x,v, t) dx dv =

∫F−1(V )

f0(y,u) dy du.

• Conservation of momentum

(30)d

dt

∫vf dxdv =

d

dt

∫J dx = 0.

• Conservation of energy

(31)d

dt

[1

2

∫v2f dxdv +

1

2

∫E2 dx

]= 0.

Proof. The system defining the associated characteristics writes

dX

dt= V(t),(32)

dV

dt= −E(X(t), t).(33)

We denote by (X(t;x,v, s),V(t;x,v, s)), or more concisely (X(t),V(t)) when the depen-dency with respect to the initial conditions is not explicitly needed, the unique solutionat time t of this system which takes the value (x,v) at time s.

Using (32)-(33), the Vlasov equation (24) can be expressed equivalently

d

dt(f(X(t),V(t)) = 0.

We thus have

f(x,v, t) = f0(X(0;x,v, t),V(0;x,v)).

From this expression, we deduce that f verifies a maximum principle which can bewritten as f0 is non negative

0 ≤ f(x,v, t) ≤ max(x,v)

(f0(x, v)).

Multiplying the Vlasov equation by (24) par fp−1 and integrating on the wholephase-space we obtain

d

dt

(∫(f(x,v, t))p dx dv

)= 0,

so that the Lp norms of f are conserved for all p ∈ N∗. Let us notice that the L∞ is alsoconserved thanks to the maximum principle (27).

Integrating on a arbitrary volume V ol of phase space and using that f is conservedalong the characteristics we get∫

V olf(x,v, t) dx dv =

∫V ol

f(X(t;x,v, t),V(t;x,v, t), t) dx dv

=

∫V ol

f(X(0;x,v, t),V(0;x,v, t), 0) dx dv

=

∫V ol

f0(X(0;x,v, t),V(0;x,v, t) dx dv,

2. THE VLASOV-POISSON SYSTEM 23

now making the change of variables (y,u) = F(x,v) defined by y = X(0;x,v, t),u =V(0;x,v, t) whose jacobian is equal to 1 thanks to proposition 4 as

∇(x,v) ·(

v−E

)= 0,

we obtain ∫V ol

f(x,v, t) dx dv =

∫F−1(V ol)

f0(y,u) dy du.

Let us now proceed to the conservation of momentum (or total current density). Weshall use the following equality that is verified for any vector u depending on x in aperiodic domain

(34)

∫(∇× u)× u dx = −

∫ (u(∇ · u) +

1

2∇u2

)dx = −

∫u(∇ · u) dx.

Let us notice in particular that taking u = E in the previous equality with E solutionof the Poisson equation (25), we get, as ∇ × E = 0 and ∇ · E = −∆φ = 1 − ρ, that∫E(1 − ρ) dx = 0. As moreover E = −∇φ and as we integrate on a periodical domain∫E dx = 0. It results that

(35)

∫Eρ dx = 0.

Let us now introduce the Green formula on the divergence:

(36)

∫Ω∇ · Fq +

∫ΩF · ∇q =

∫∂Ω

(F · n) q ∀F ∈ H(div,Ω), q ∈ H1(Ω),

where classically H1(Ω) is the subset of L2(Ω) the square integrable functions, of thefunctions whose gradient is in L2(Ω); andH(div,Ω) is the subset of L2(Ω) of the functionswhose divergence is in L2(Ω).

Let’s multiply the Vlasov equation (24) by v and integrate in x and in v

d

dt

∫vf dxdv +

∫∇x · (v ⊗ vf) dxdv −

∫v∇v · (Ef) dxdv = 0.

The second integral vanishes as the domain is periodic in x and the Green formula onthe divergence (36) gives for the last integral

−∫

v∇v · (Ef) dxdv =

∫Ef dxdv =

∫Eρ dx = 0,

using (35). It finally follows that

d

dt

∫vf dxdv =

d

dt

∫J dx = 0.

In order to obtain the energy conservation property, we start by multiplying theVlasov equation by v · v = |v|2 and we integrate on phase space

d

dt

∫|v|2f dxdv +

∫∇x · (|v|2vf) dxdv −

∫|v|2∇v · (Ef) dxdv = 0.

As f is periodic in x, we get, integrating in x that∫∇x · (|v|2vf) dxdv = 0

and the Green formula on the divergence (36) yields∫|v|2∇v ·E dxdv = −2

∫v · (Ef) dxdv = −2

∫E · J dx.


So

(37)d

dt

∫|v|2f dxdv = −2

∫E · J dx = 2

∫∇φ · J dx.

On the other hand, integrating the Vlasov equation (24) with respect to v, we get the

charge conservation equation, generally called continuity equation: ∂ρ∂t + ∇ · J = 0.

Then, using again the Green formula (36), the Poisson equation (25) and the continuityequation, we obtain∫

∇φ · J dx =

∫φ∇ · J dx = −

∫φ∂ρ

∂tdx =

∫φ∂∆φ

∂tdx = −1

2

d

dt

∫∇φ · ∇φdx.

And so, plugging this equation in (37) and using that E = −∇φ, we get the conservationof energy.

CHAPTER 4

Numerical methods for the Vlasov equation

1. Operator splitting

In the Vlasov equation without a magnetic field, the advection field in x, which is v,does not depend on x and the advection field in v, which is E(x, t), does not depend onx. Therefore it is often convenient to decompose these two parts, using the techniquecalled operator splitting.

Let us consider the non relativistic Vlasov-Poisson equation which reads

∂f

∂t+ v · ∇x f +

q

mE · ∇v f = 0,

coupled with the Poisson equation −∆φ = 1 − ρ(t,x) = 1 −∫f(t,x,v) dv, E(x, t) =

−∇φ. Throught this coupling, E depends on f , which makes the Vlasov-Poisson systemnon linear.

We shall split the equation into the following two pieces:

(38)∂f

∂t+ v · ∇x f = 0,

with v fixed and

(39)∂f

∂t+

q

mE(x, t) · ∇v f = 0,

with x fixed. We then get two constant coefficient advections that can are easier tosolve. This is obvious for (38) as v does not depend on t and x. On the other hand,

integrating (39) with respect to v, we get that ∂ρ∂t = ∂

∂t

∫f(t,x,v) dv = 0, so that ρ and

consequently E does not change when this equation is advanced in time. So that E(t, x)needs to be computed with the initial f for this equation and does then depend neitheron t, nor x.

Remark 4. When the starting equation has some features which are important forthe quality of the numerical solution, it is essential not to remove then when doingoperator splitting. In particular, if the initial equation is conservative, it is generally agood idea to split such that each of the split equation is conservative.

In order to analyze the error resulting from operator splitting, let us consider thefollowing system of equations

(40)du

dt= (A+B)u,

where A and B are any two differential operators (in space), that are assumed constantbetween tn and tn+1. The formal solution of this equation on one time step reads:

u(t+ ∆t) = e∆t(A+B)u(t).

25

26 4. NUMERICAL METHODS FOR THE VLASOV EQUATION

Let us split the equation (40) into

du

dt= Au,(41)

du

dt= Bu.(42)

The formal solutions of these equations taken separately are

u(t+ ∆t) = e∆tAu(t) and u(t+ ∆t) = e∆tBu(t).

The standard operator splitting method consists in solving successively on one timestep first (41) and then (42). Then one gets on one time step

u(t+ ∆t) = e∆tBe∆tAu(t).

If the operators A and B commute e∆tBe∆tA = e∆t(A+B) and the splitting is exact. Thisis the case in particular when considering a constant coefficient advection equation ofthe form

∂u

∂t+ a

∂u

∂x+ b

∂u

∂y= 0.

This can be checked using the method of characteristics. Note that such an equation isalso a good first test case to validate a Vlasov code.

In the case when A and B do not commute, the splitting error can be decreased bysolving first (41) on a half time step, and then (42) on a full time step and again (41) ona half time step. This method is known as the Strang splitting method. It correspondsto the formal solution

u(t+ ∆t) = e∆t2Ae∆tBe

∆t2Au(t).

The error committed at each time step by the operator splitting method when theoperators do not commute is given by

Proposition 6. • The standard splitting method is of order 1 in time.• The Strang splitting method is of order 2 in time.

Proof. In order to find the error we need to expand the matrix exponential. Onthe one hand we have

e∆t(A+B) = I + ∆t(A+B) +∆t2

2(A+B)2 +O(∆t3),

and on the other hand

e∆tBe∆tA = (I + ∆tB +∆t2

2B2 +O(∆t3))(I + ∆tA+

∆t2

2A2 +O(∆t3))

= I + ∆t(A+B)∆t2

2(A2 +B2 + 2BA) +O(∆t3)).

But as A and B do not commute, we have (A+B)2 = A2 +AB +BA+B2. It follows

that e∆t(A+B)− e∆tBe∆tA = O(∆t2), which leads to a local error of order 2 and a globalerror of order 1.

For the Strang splitting method, we have

e∆t2Ae∆tBe

∆t2A =(I +

∆t

2A+

∆t2

4A2 +O(∆t3)))(I + ∆tB +

∆t2

2B2 +O(∆t3))

(I +∆t

2A+

∆t2

4A2 +O(∆t3)))

=I + ∆t(A+B) +∆t2

2(A2 +B2 +BA+AB) +O(∆t3).

2. INTERPOLATION 27

We thus obtain a local error of order 3 and thus a global error of order 2 for the methodof Strang.

Remark 5. It is possible to obtain splitting methods of order as high as desiredby taking adequate compositions of the two operators. Details on high order splittingmethods can be found in [44].

Remark 6. The Strang splitting method can also be generalized to more that twooperators. If A = A1 + · · ·+An, the following decomposition will be of global order 2:

e∆t2A1 . . . e

∆t2An−1e∆tAne

∆t2An−1 . . . e

∆t2A1 .

2. Interpolation

One of the main buiding blocks of a semi-Lagrangian method is interpolation. Inorder for the method not to be too diffusive it is important to use a good interpolationmethod which is accurate enough. Typically, linear interpolation is way too diffusive.The method of choice in many semi-Lagrangian codes is cubic splines which proves veryrobust and accurate.

2.1. Splines.2.1.1. General definition. Consider a 1D grid of an interval [a, b] a = x1 < x2 < ... <

xN = b. We define Sp(a, b) the linear space of splines of degree p on [a, b] by

Sp(a, b)S ∈ Cp−1([a, b]) | S|[xi,xi+1] ∈ Pp([xi, xi+1]),

where Pp([xi, xi+1]) denotes the space of polynomials of degree p on [xi, xi+1], which isof dimension is p+ 1.

Let us first consider periodic splines which are very usefull for our applications as thespecial case. In this case we consider periodic functions of period b− a. These functionstake the same value at a and b so that x1 and xN can be considered the same to bethe same grid point. Let us compute the dimention of Sp in this case. Sp is includedin the space of N − 1 piecewise polynomials of dimension (N − 1)(p+ 1). Its dimensionis reduced by the continuity requirements on the spline and its derivatives at each gridpoint, which is altogether (N − 1)p. So that the dimension of Sp is N − 1 for periodicsplines. In this case a spline can be determined by an interpolation condition at eachgrid point.

Now on a regular interval this is slightly modified. Indeed the spline is still a subspaceof the space of N − 1 piecewise polynomials of dimension (N − 1)(p + 1), but now thecontinuity requirements are only at the N − 2 interior points. So that the dimension ofSp in this case is (N − 1)(p + 1) − (N − 2)p = N + p − 1. This is larger than N forp ≥ 2 so that boundary conditions are needed in addition to the interpolation conditionsat the grid points to uniquely determine the spline. A classical boundary condition isHermite boundary conditions, which state that all derivatives up to order (p− 1)/2 aregiven at each of the two boundaries of the interval for odd degree splines that are usedin practise for interpolation.

Following the arguments to compute the dimension of the spline space, a natural wayof computing the spline would be to compute the local polynomials api x

p + ap−1i xp−1 +

· · · + a1ix + a0

i on each interval i, 1 ≤ i ≤ N − 1, using the interpolation values at thegrid points and the continuity relations.

This can be used in practise to compute spline interpolations but it is in generalmore efficient to use a set of basis functions called B − splines.


2.1.2. B-splines. We define B − splines as follows: Let T = (ti)16i6N+k be a non-decreasing sequence points. In the splines jargon these points are called knots. This ismore general than standard spline interpolation as considered previously. In particularrepeated knots can be considered.

Definition 2 (B-Spline). The i-th B-Spline of degree p is defined by the recurrencerelation:

(43) Np+1j = wp+1

j Npj + (1− wp+1

j+1)Npj+1

where,

wp+1j (x) =

x− tjtj+p − tj

N0j (x) = χ[tj ,tj+1[(x)

We note some important properties of a B-splines basis:

• B-splines are piecewise polynomial of degree p.• Positivity: Np

j (x) ≥ 0 for all x.

• Compact support; the support of Np+1j is contained in [tj , .., tj+k].

• Partition of unity :∑N

i=1Npi (x) = 1,∀x ∈ R

• Local linear independence.• If a knot t has a multiplicity m then the B-spline is C(p−m) at t.

The derivative of a B-spline of degree p can be computed as a simple difference ofB-splines of degree p− 1

(44) Npi′(x) = p

(Np−1i (x)

ti+p − ti−

Np−1i+1 (x)

ti+p+1 − ti+1

).

An important special case is the case of uniformly spaced knots on an infinite orperiodic grid. In this case the splines are often called cardinal splines and all the B-splines are translates of each other, so that the basis can be defined with only oneelement denoted by Np for the degree p, if h is the spacing between successive knotsthen the full basis is composed of the translates (Np(· − jh))j∈I , where the index setis I = Z or a finite subset of Z in the periodic case. The cardinal splines are generallydefined with the integers as knots. In this case formula (43) becomes

(45) Np+1(x) =x

pNp(x) +

p+ 1− xp

Np(x− 1),

with N0(x) = 1 if x ∈ [0, 1[ and 0 else. It easily follows that the support of Np is[0, p+ 1].

Remark 7. B-Splines Nph on the uniform grid jh, j ∈ Z verify Np

h(x) = Np(x/h).It is thus sufficient to define B-splines on integer knots.

Note that in addition to the general properties of the splines, the cardinal splinesalso verify

Np+1(x) =

∫ 1

0Np(x− t) dt

and

Np(p+ 1

2+ x) = Np(

p+ 1

2− x) ∀x ∈ R.

See the book of Chui [15] for proofs of these properties and more on cardinal splines.Using this properties we can prove the following lemma on the first moment of a

cardinal spline. Similar properties can also be obtained for higher order moments [39].

2. INTERPOLATION 29

Lemma 1. For all x ∈ R, if Np is the cardinal spline of degree p we have∑j

(j − x)Np(j − x) =

∫ p+1

0tNp(t) dt =: Mp.

In other words, the sum is independent on x and is equal to the moment of the cardinalspline that we denote by Mp.

Proof. Let us first denote for any given p Mp(x) =∑

j(j − x)Np(j − x). Using

(45) we have

p∑j

(j − x)Np+1(j − x) =∑j

(j − x)2Np(j − x) +∑j

(j − x)(p+ 1− j + x)Np(j − x− 1)

=∑j

(j − x)2Np(j − x) +∑j

(j + 1− x)(p− j + x)Np(j − x)

making a change of index in the last sum. Then combining both sums

p∑j

(j − x)Np+1(j − x) = p∑j

Np(j − x) + (p− 1)∑j

(j − x)Np(j − x)

= p+ (p− 1)∑j

(j − x)Np(j − x),

due to the partition of unity property. So that we get the recurrence relation

Mp+1(x) = 1 +p− 1

pMp(x).

For p = 1, M1(x) involves only two non vanishing terms. Denoting by bxc the floor ofx, i.e. the greatest integer smaller than x, only the terms corresponding to j = bxc+ 1and j = bxc+ 2 do not vanish in the sum. Then denoting by α = x− bxc the fractionalpart of x, we have 0 ≤ α ≤ 1 and

M1(x) = (1− α)N1(1− α) + (2− α)N1(2− α) = (1− α)2 + (2− α)α = 1,

as N1(x) = x on [0, 1] and N1(x) = 2 − x on [1, 2]. So M1(x) does not depend on xand then by induction using the recurrence relation previously derived Mp(x) does notdepend on x, only on p.

On the other hand let us directly compute Mp =∫ p+1

0 tNp(t) dt. Using (45) we get

pMp+1 =

∫ p+2

0xNp+1(x) dt

=

∫ p+1

0x2Np(x) dx+

∫ p+2

1(p+ 1− x)xNp(x− 1) dx

=

∫ p+1

0x2Np(x) dx+

∫ p+1

0(p− x)(x+ 1)Np(x) dx

=

∫ p+1

0(x2 + (p− x)(x+ 1))Np(x) dx

=

∫ p+1

0(p+ (p− 1)x)Np(x) dx

= p+ (p− 1)Mp,

using that∫ p+1

0 Np(x) dx = 1 and the definition of Mp, we hence get the same recurrence

formula for Mp as we had for Mp(x) it thus remains to check that M1 = 1 to conclude.


0 1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

Figure 1. Cubic splines with open boundary conditions with knots at integers

For this a straightforward computation yields

M1 =

∫ 2

0xN1(x) dx =

∫ 1

0x2 dx+

∫ 2

1x(2− x) dx =

1

3+

2

3= 1.

2.2. Using B-splines for spline interpolation. The B-splines form a basis ofthe spline space SpN . In the case of a periodic domain, we saw that the dimension of SpNis exactly the number of grid points. In this case the knots can be taken to be exactlythe grid points. The situation is a little bit more complicated for a bounded interval, inwhich case it more knots than grid points are needed to define the B-splines that willgenerate SpN . Two natural possibilities exist, the first one is to replicate the knots at thetwo extremities of the interval. This has the advantage that the spline is interpolatoryat the boundary so that Dirichlet boundaries are easily handled. On the other hand,this solution has the drawback that the shape of the B-splines changes a both ends ofthe domain which might be unwanted in particular if the grid is uniform so that theshape of the splines is the same for all the inner splines. In this case, another option, isto mirror the points close to the boundary.

In any case let us denote M = dimSpN . Recall that M = N − 1 for periodic splinesand M = N + p − 1 for bounded splines. Then a spline S ∈ SpN can be written, usingthe B-spline basis

S(x) =M∑i=1

ciNpi (x).

2. INTERPOLATION 31

In order to use this formula for interpolation we first need to determine the splinecoefficients (ci)1≤i≤M . These can be determined by the interpolation conditions S(xk) =f(xk) at the grid points and the boundary conditions in the case of non periodic splines.

2.3. Cubic spline interpolation. Let us consider a uniform mesh of the interval[a, b] defined by xi = a + ih, i = 0, . . . , N , with h = b−a

N . Let f ∈ Ck([a, b]), k ≥ 0. Itscubic spline interpolant fh on this mesh is defined by fh(xi) = f(xi) for i = 0, . . . , N ,fh ∈ P3([xi, xi+1]) and fh ∈ C2([a, b]).

In the case of a periodic domain, i.e., if [a, b] corresponds to one period of theperiodic functions f and fh, these conditions are sufficient to determine uniquely fh. Elseboundary conditions are needed, often Hermite type boundary conditions, consisting ingiving the values of f ′h(a) and of f ′h(b) at the ends of the interval or the so-called naturalboundary conditions, consisting in setting f ′′h (a) = f ′′h (b) = 0 are used.

It is convenient to have an expression of fh using the cubic B-splines basis, whichare the translations of the function S3. Let us recall the expression of S3 on our mesh.

S3(x) =1

6

(2− |x|h )3 if h ≤ |x| < 2h,

4− 6(xh

)2+ 3

(|x|h

)3if 0 ≤ |x| < h,

0 else.

Let us first deal with the periodic case. We assume that all functions we consider are

periodic of period b − a. Then in particular f(p)h (a) = f

(p)h (b) for p = 0, 1, 2. The point

xN of the mesh corresponds to the point x0 and no additional value of the unknown isdefined there.

The expression of fh on the B-splines basis then reads

fh(x) =N−1∑j=0

αjS3(x− xj),

and the coefficients αi are determined by the interpolation conditions.

f(xi) = fh(xi) =N−1∑j=0

αjS3(xi − xj).

But S3(xi − xi) = 23 , S3(xi − xi+1) = S3(xi − xi−1) = 1

6 and S3(xi − xj) = 0 if|xi − xj | ≥ 2h.

We thus get a linear system with unknowns αi, i = 0, N − 1 :

αi−1 + 4αi + αi+1 = 6f(xi), 0 ≤ i ≤ N − 1,

with because of periodicity α−1 = αN−1 and αN = α0. This system can be written inmatrix form Aα = b with

A =

4 1 0 . . . 0 11 4 1 0 . . . 0

0. . .

. . .. . .

. . ....

.... . .

. . .. . .

. . . 00 . . . 0 1 4 11 0 . . . 0 1 4

, α =

α0...

αN−1

, b = 6h

f(x0)...

f(xN−1)

.

As the matrix A is strictly diagonally dominant it is non singular and allows to determinethe unknows α and hence the function fh uniquely.

2.4. Discrete Fourier Transform.


2.4.1. Definition. Let P be the symmetric matrix formed with the powers of the

nth roots of unity the coefficients of which are given by Pjk = 1√ne

2iπjkn . Denoting by

ωn = e2iπn , we have Pjk = 1√

nωjkn .

Notice that the columns of P , denoted by Pi, 0 ≤ i ≤ n − 1 are the vectors Xi

normalized so that P ∗i Pj = δi,j . On the other hand the vector Xk corresponds to a

discretization of the function x 7→ e−2iπkx at the grid points xj = j/n of the interval[0, 1]. So the expression of a periodic function in the base of the vectors Xk is thusnaturelly associated to the Fourier series of a periodic function.

Definition 3. Discrete Fourier Transform.

• The dicrete Fourier transform of a vector x ∈ Cn is the vector y = P ∗x.• La inverse discrete Fourier transform of a vector y ∈ Cn is the vectorx = P ∗−1x = Px.

Lemma 2. The matrix P is unitary and symmetric, i.e. P−1 = P ∗ = P .

Proof. We clearly have P T = P , so P ∗ = P . There remains to prove that PP = I.But we have

(PP )jk =1

n

n−1∑l=0

ωjlω−lk =1

n

n−1∑l=0

e2iπnl(j−k) =

1

n

1− e2iπnn(j−k)

1− e2iπn

(j−k),

and so (PP )jk = 0 si j 6= k and (PP )jk = 1 if j = k.

Corollary 1. Let F,G ∈ Cn and denote by F = P ∗F and G = P ∗G, their discreteFourier transforms. Then we have

• the discrete Parseval identity:

(46) (F,G) = F T G = F T¯G = (F , G),

• The discrete Plancherel identity:

(47) ‖F‖ = ‖F‖,

where (., .) and ‖.‖ denote the usual euclidian dot product and norm in Cn.

Proof. The dot product in Cn of F = (f1, . . . , gn)T and G = (g1, . . . , gn)T is definedby

(F,G) =N∑i=1

figi = F T G.

The using the definition of the inverse discrete Fourier transform, we have F = PF ,G = PG, we get

F T G = (PF )TPG = F TP T P¯G = F T

¯G,

as P T = P and P = P−1. The Plancherel identity follows from the Parseval identity bytaking G = F .

Remark 8. The discrete Fourier transform is defined as a matrix-vector multipli-cation. Its computation hence requires a priori n2 multiplications and additions. Butbecause of the specific structure of the matrix there exists a very fast algorithm, calledFast Fourier Transform (FFT) for performing it in O(n log n) operations. This makesit particularly interesting for many applications, and many fast PDE solvers make useof it.

2. INTERPOLATION 33

2.4.2. Approximation of the coefficients of a Fourier series using the FFT. A L−periodicfunction f can be expressed by its Fourier series. More precisely the classical Dirich-let theorem states that if f is C1 its Fourier series converges uniformly towards f(x)pointwise for any x, where the Fourier series is defined by

f(x) =+∞∑

k=−∞fke

i k2πLx,

where the Fourier coefficients ck are defined by

fk =1

L

∫ L

0f(x)e−i

k2πLx dx.

In order to compute a numerical approximation of the Fourier coeffients, we definea mesh with N points on one period [0, L[ such that xj = jL/N , 0 ≤ j ≤ N − 1. Wedenote by fj = f(xj). We then have

fk =1

L

N−1∑j=0

∫ xj+1

xj

f(x)e−ik2πLx dx.

As f is known only at the grid points the integral is approximated by the trapezoidalrule on each grid interval. Note that due to the Euler-MacLaurin formula the composedtrapezoidal rule is very accurate for periodic function. More precisely, if f ∈ C2p([0, L])and L−periodic, the composed trapezoidal rule is of order 2p. We then get

fk ≈1

L

L

N

N∑j=0

fje−i k2π

LjLN =

1

N

N∑j=0

fje−i k2πj

N .

Such a numerical approximation involves a sampling of the intial function at N pointsxj . Because of this sampling some information on the initial function is lost and the

Fourier series only contains N distinct values fk. Indeed, for any k ∈ Z we have

fk+N =1

N

N∑j=0

fje−i (k+N)2πj

N = fk.

These N distinct values approximate fk for −N/2 ≤ k ≤ N/2 − 1. The other modesare not represented by the discrete Fourier tranform. Notice that the correspondingfrequencies ω = 2πk

L lie in the interval [−π/L, π/L[.

Note that the discrete Fourier transform gives fk+N for 0 ≤ k ≤ N − 1. In orderto use it for approximating Fourier series, we use the N-periodicity of the coefficients todefine fk pour −N/2 ≤ k < 0 from fk for N/2 ≤ k ≤ N−1. Matlab and other numericalsoftware provide the funciton fftshift to transfer the N/2− 1 last modes provided bythe FFT to the beginning of the array.

2.4.3. Computing an approximate solution of the Poisson equation using the FFT.For the approximationg a linear PDE with constant coefficients on a periodic domainthe FFT is the simplest and often fastest method. If the solution is smooth it providesmoreover spectral convergence, which means that it converges faster than a polynomialapproximation of any order, so that very good accuracy can be obtained with relativelyfew points. The exact number depends of course on the variation of the solution. Letus explain how this works for the Poisson equation on a periodic domain that we shallneed for our simulations.


Consider the Poisson equation −∆φ = ρ on a periodic domain of R3 of periodL1, L2, L3 in each direction. The solution is uniquely defined provided we assume thatthe integral of φ on one period vanishes.

We look for an approximation of φ of the form in the form of a truncated Fourierseries

φh(x1, x2, x3) =

N1/2−1∑k1=−N1/2

N2/2−1∑k2=−N2/2

N3/2−1∑k3=−N3/2

φk1,k2,k3eik·x,

where we denote by k = (2πk1/L1, 2πk2/L2, 2πk3/L3) and by x = (x1, x2, x3).

Remark 9. Note that in principle, it would be natural to truncate the Fourier seriesin a symmetric way around the origin, i.e. from −N/2 to N/2. However, because theFFT is most efficient when the number of points is a power of 2, we need to use aneven number of points which leads to the truncation we use here. See Canuto, Hussaini,Quarteroni and Zang for more details [11].

We assume the same decomposition for ρh. Then taking explicitly the Laplace of φhwe get

−∆φh(x1, x2, x3) =

N1/2−1∑k1=−N1/2

N2/2−1∑k2=−N2/2

N3/2−1∑k3=−N3/2

|k|2φk1,k2,k3eik·x.

Then using the collocation principle, we identify this expression with that of ρh at thediscretisation points j = (j1L1/N1, j2L2/N2, j3L3/N3) with 0 ≤ ji ≤ Ni − 1:∑

k1,k2,k3

|k|2φk1,k2,k3eik·j =

∑k1,k2,k3

ρk1,k2,k3eik·j.

Then as the (eik·j)(k1,k2,k3) form a basis of RN1 ×RN2 ×RN2 we can identify the coeffi-cients, so that we have a simple expression of the Fourier coefficients of φh with respectto those of ρh for |k| 6= 0:

φk1,k2,k3 =ρk1,k2,k3

|k|2, −Ni/2 ≤ ki ≤ Ni/2− 1,

and because we have assume that the integral of φ is 0, we have in addition φ0,0,0 = 0.Now, to complete the algorithm , we shall describe how these coefficients can be

computed from the grid values by a 3D discrete Fourier transform.In 1D, φk is defined by φk = 1√

N

∑N−1i=0 φie

2iπjk/N from which it is easy to see

that φk+lN = φk for any integer l. So after having computed φk using a Fast Fourier

Transform for 0 ≤ k ≤ N − 1 we get the negative coefficients φk for −N/2 ≤ k ≤ −1

from the known coefficients by the relation φk = φk+N . Finally to go to the 3D case,it is enough to see that the 3D discrete Fourier transform is nothing but a series of 1Dtransforms in each direction.

Remark 10. In order to compute the electric field E = −∇φ in the pseudo-spectralapproximation, we just multiply each mode φk1,k2,k3 by the corresponding ik. Bewarehowever, that because we use an unsymmetric truncated Fourier series, the mode −N/2of the electric field needs to be set to 0 in order to get back a real electric field by inverseFourier transform. Indeed for a real field (uj)0≤j≤N−1, the corresponding N/2 mode is∑N−1

j=0 (−1)juj which is real. Hence, as ρ and then φ are real, their corresponding N/2mode is real, and thus the same mode for E would be purely imaginary and not realunless it is 0. Note that setting this mode to 0 introduces an additional error of theorder of truncation error of the series, and thus is acceptable.

2. INTERPOLATION 35

2.4.4. Circulant matrices.

Definition 4. A matrix of the form

M =

c0 c1 c2 . . . cn−1

cn−1 c0 c1 cn−2

cn−2 cn−1 c0 cn−3...

. . ....

c1 c2 c3 . . . c0

with c0, c1, . . . , cn−1 ∈ R is called circulant.

Proposition 7. The eigenvalues of the circulant matrix M are given by

(48) λk =

n−1∑j=0

cjωjk,

where ω = e2iπ/n.

Proof. Let J be the circulant matrix obtained from M by taking c1 = 1 and cj = 0for j 6= 1. We notice that M can be written as a polynomial in J

M =n−1∑j=0

cjJj .

As Jn = I, the eigenvalues of J are the n-th roots of unity that are given by ωk = e2ikπ/n.Looking for Xk such that JXk = ωkXk we find that an eigenvector associated to theeigenvalue λk is

Xk =

1ωk

ω2k

...

ω(n−1)k

.

We then have that

MXk =n−1∑j=0

cjJjXk =

n−1∑j=0

cjωjkXk,

and so the eigenvalues of M associated to the eigenvectors Xk are

λk =

n−1∑j=0

cjωjk.

Proposition 8. Any circulant matrix C can be written in the form C = PΛP ∗

where P is the matrix of the discrete Fourier transform and Λ is the diagonal matrixof the eigenvalues of C. In particular all circulant matrices have the same eigenvectors(which are the columns of P ), and any matrix of the form PΛP ∗ is circulant.

Corollary 2. We have the following properties:

• The product of two circulant matrix is circulant matrix.• A circulant matrix the eigenvalues of which are all non vanishing is invertible

and its inverse is circulant.


Proof. The key point is that all circulant matrices can be diagonalized in the samebasis of eigenvectors. If C1 and C2 are two circulant matrices, we have C1 = PΛ1P

∗ andC2 = PΛ2P

∗ so C1C2 = PΛ1Λ2P∗.

If all eigenvalues of C = PΛP ∗ are non vanishing, Λ−1 is well defined and PΛP ∗PΛ−1P ∗ =I. So the inverse of C is the circulant matrix PΛ−1P ∗.

Because of the fact that all circulant matrices diagonalise in the Fourier basis, theFFT provides a fast and convenient tool for solving linear systems or performing productsinvolving circulant matrices. In particular performing spline interpolation at a constantdisplacement from each grid point can be writen in matrix form as: MC = Fn (computespline coefficients from point values), then Fn+1 = DC where M and D are circulantmatrices. Combining both relations we obtain Fn+1 = DM−1Fn. Denoting λD and λMthe diagonal matrices of the eigenvalues of respectively D and M that can be computingeasily with formula (48) using the coefficients of the circulant matrix. Indeed we have

Fn+1 = PΛDλ−1M P ∗Fn,

And multiplying by P ∗ consists in performing a Discrete Fourier Tranform. So that thealgorithm for the computation becomes:

(1) Fn = FFT (Fn),

(2) Gnj = FnλD,j/λM,j for i = 0, n− 1,

(3) Fn+1 = iFFT (Gn), where iFFT denotes an inverse FFT.

2.4.5. Lagrange interpolation. Lagrange interpolation, although dissipative at loworder can be a good alternative to spline interpolation if a high enough order is used.In pratice odd degree Lagrange interpolation starting from degree 7 gives quite goodresults.

Let us recall how this can be implemented efficiently at arbitrary order. The La-grange interpolation polynomial of degree N at points x0, . . . , xN of a smooth functionf is defined by

p(x) =N∑j=0

fjlj(x),

where fj = f(xj) and lj(x) is he jth Lagrange polynomial of degree N uniquely definedby lj(xi) = δij , δij being the Kronecker symbol which is 1 if i = j and 0 else. The explicitformula for lj(x) is

lj(x) =

N∏i=0,i 6=j

(x− xi)

N∏i=0,i 6=j

(xj − xi)

.

Computing the Lagrange polynomials for Lagrange interpolation is not very convenientas it involves O(n) multiplications and sums for each point to be interpolated and needsto be started anew when an interpolation point is added. In order to simplify this weintroduce the function

ω(x) =N∏i=0

(x− xi), and wj =1

ω′(x)=

1N∏

i=0,i 6=j(xj − xi)

.

3. SEMI-LAGRANGIAN METHODS 37

The the Lagrange interpolating polynomial can be written

p(x) = ω(x)(N∑j=0

fjwj

x− xj).

And as the interpolation is exact for f = 1, we get an expression for ω(x)

1 = ω(x)(

N∑j=0

wjx− xj

),

so that we get the following simple formula that is convenient and efficient for Lagrangeinterpolation as the coefficients wj need to be computed only once for all interpolationpoints:

p(x) =

∑Nj=0 fj

wjx−xj∑N

j=0wjx−xj

.

This is called the barycentric interpolation formula. See the review article by Beirut andTrefethen [6] for further information.

3. Semi-Lagrangian methods

Semi-Lagrangian methods have become, far behing the Particle-In-Cell (PIC) methoda classical choice for the numerical solution of the Vlasov equation, thanks to their goodprecision and their lack of numerical noise as opposite to PIC methods. They need aphase space mesh and thus are very computationally intensive when going to higherdimensions. Indeed a 3D simulation requires a 6D mesh of phase space. For this reason,semi-Lagrangian methods have become very popular for 1D or 2D problems, but thereare still relatively few 3D simulations being performed with this kind of method.

The specificity of semi-Lagrangian methods, compared to classical methods for nu-merically solving PDEs on a mesh, is that they use the characteristics of the scalarhyperbolic equation, along with an interpolation method, to update the unknown fromone time step to the next. These semi-Lagrangian methods exist in different varieties:backward, forward, point based or cell based.

Most semi-Lagrangian solvers are based on cubic spline interpolation which hasproven very efficient in this context. So let us start by introducing this interpolation.

3.1. The classical semi-Lagrangian method. Let us consider an abstract scalaradvection equation of the form

(49)∂f

∂t+ a(x, t) · ∇f = 0.

The characteristic curves associated to this equation are the solutions of the ordinarydifferential equations

dX

dt= a(X(t), t).

We shall denote by X(t,x, s) the unique solution of this equation associated to the initialcondition X(s) = x.

The classical semi-Lagrangian method is based on a backtracking of characteristics.Two steps are needed to update the distribution function fn+1 at tn+1 from its value fn

at time tn :

(1) For each grid point xi compute X(tn;xi, tn+1) the value of the characteristic attn which takes the value xi at tn+1.


(2) As the distribution solution of equation (49) verifies

fn+1(xi) = fn(X(tn;xi, tn+1)),

we obtain the desired value of fn+1(xi) by computing fn(X(tn;xi, tn+1) byinterpolation as X(tn;xi, tn+1) is in general not a grid point.

These operations are represented on Figure 2.

Figure 2. Sketch of the classical semi-Lagrangian method.

Remark 11. This semi-Lagrangian method is very diffusive if a low order (typicallylinear) interpolation is used. In practice one often used cubic splines or cubic Hermiteinterpolation, which offer a good compromise between accuracy and efficiency.

This semi-Lagrangian method has been initially introduced for the 1D Vlasov-Poissonequation by Cheng and Knorr [14] in 1976. It was based on a cubic spline interpolationand a directional Strang splitting method and is still one of the most used methods forthis problem.

Let us now specify the algorithm for the 1D Vlasov-Poisson problem where theunknown is the distribution function for the electrons and in presence of motionlessneutralizing background ions on a domain [0, L] periodic in x and infinite in v. Theequations then read

∂f

∂t+ v

∂f

∂x− E(x, t)

∂f

∂v= 0,

dE

dx= ρ(x, t) = 1−

∫f(x, v, t) dv,

with the initial condition f(x, v, 0) = f0(x, v), verifying∫f0(x, v) dx dv = L.

The infinite velocity space is truncated to a segment [−A,A] sufficiently large so thatf stays of the order of the round off errors for velocities less than −A or larger thanA during the whole simulation (in practise in the normalized examples we are going toconsider, taking A of the order of 10 is very safe for all our test cases. Let us define auniform grid of phase space xi = iL/N , i = 0, . . . , N−1 (the point xN which correspondsto x0 is not used), vj = −A+ j2A/M , j = 0, . . . ,M .

The full algorithm can in this case be written:

(1) Initialization. Assume the initial distribution function f0(x,v) given. Wededuce ρ(x, 0) = 1 −

∫f0(x, v) dv, and then compute the initial electric field

E(x, 0) solving the Poisson equation.


(2) Update from tn to tn+1. The function fn is known at all grid points (xi, vj)of phase space and En is known at all grid points xi of the configuration space.• We compute f∗ by solving

∂f

∂t+ En

∂f

∂v= 0

on a half time step ∆t2 using the semi-Lagrangian method.

• We compute f∗∗ by solving on a full time step

∂f

∂t+ v

∂f

∂x= 0

from the initial condition f∗.• We compute ρn+1(x) = 1−

∫f∗∗(x, v) dv and then the corresponding elec-

tric field En+1 using the Poisson equation.• We compute fn+1 by solving on a half time step

(50)∂f

∂t+ En+1∂f

∂v= 0

from the initial condition f∗∗.Note that the actual ρn+1 can be computed using f∗∗(x, v) (instead of fn+1(x, v)),as the charge density corresponding to f∗∗(x, v) is identical to that associatedto fn+1(x, v). Indeed, we go from f∗∗(x, v) to fn+1(x, v) by solving (50), andwe notice, integrating this equation in v that it implies that d

dt

∫f(x, v, t) dv = 0

and so that ρ is not modified during this stage.

3.2. Conservation properties of the split semi-Lagrangian method withB-splines on a uniform mesh. Let us prove here the exact conservation propertiesof the classical semi-Lagrangian scheme on a uniform mesh that we just introduced.We shall check that during each split step of the method, which is a constant coeffi-cient advection, total mass and momentum are exactly conserved in the case of periodicboundary conditions in x and infinite domain in v. These exact conservation propertieswill be violated by the truncation performed in the velocity domain. However if vminand vmax are taken large enough this will be of the order of the round-off error and hasno influence on the scheme.

3.2.1. Conservation of mass.

Proposition 9. The discrete mass ∆x∆v∑

i,j fi,j is exactly conserved by the nu-merical scheme.

Proof. Here we just need to check that for a constant coefficient advection the massis conserved on each line or each column. So let us consider only the 1D problem. Let usdenote by fki the value of the distribution function at the grid points at the beginning of

a split step on one given line or column and fk+1i the value of the distribution function

at the grid points at the end of the split step on then same line or column. Then if weprove that

∑i f

k+1i =

∑i f

ki , we can conclude that the total mass is conserved.

Starting from grid values fki , we first compute the spline interpolant S on a grid ofstep h (with grid points of the form xi = ih, i ∈ Z). The spline S is defined by

S(x) =∑i

ciNp(x

h− i), with fkj =

∑i

ciNp(j − i).


Note that because∫Np(x/h− i) dx = h, we have that

∫S(x) dx = h

∑ci and because

of the partition of unity property of the B-splines we also have∑j

fkj =∑i,j

ciNp(j − i) =

∑i

ci∑j

Np(j − i) =∑i

ci.

Now using the spline for interpolation at the origin of the characteristics with aconstant advection coefficient a, we get∑j

fk+1j =

∑j

S(xj−a∆t) =∑i,j

ciNp(j−i−a∆t/h) =

∑i

ci∑j

Np(j−i−a∆t/h) =∑i

ci,

using again the partition of unity property of the B-splines.It follows that

∑j f

kj =

∑i ci =

∑j f

k+1j from which we get the conservation of

discrete mass.

3.2.2. Conservation of total momentum.

Proposition 10. The discrete total momentum ∆x∆v∑

i,j fi,jvj is exactly con-

served by the numerical scheme provided the Poisson solver verifies∑

i niEi = 0 whereni = ∆v

∑j fi,j.

Proof. In this case the advection in x and v need to be treated differently. Theadvection in x is applied on lines with constant velocities, so that the same computationas the one done for the conservation of mass can be applied. Indeed, for each j we get aspreviously on the split step

∑i f

k+1i,j =

∑i f

ki,j and then multiplying by vj and summing

also on j we get the conservation of momentum for this step.The advection in v is more complex. Performing the 1D advection for each i we have

as in the previous paragraph

fk+1i,j (vj) = fk(vj + Ei∆t) =

∑l

clNp(j − l + Ei∆t/∆v).

So the new total momentum can be expressed as∑j

vjfk+1i,j (vj) =

∑l

cl∑j

j∆vNp(j − l + Ei∆t/∆v).

But∑j

jNp(j − k + Ei∆t/∆v) =∑j

(j − l + Ei∆t/∆v)Np(j − l + Ei∆t/∆v)

+ (l − Ei∆t/∆v)∑j

Np(j − l + Ei∆t/∆v),

with∑

j Np(j − l + Ei∆t/∆v) = 1 due to the partition of unity property of the splines

and due to the properties of the cardinal splines proved in Lemma 1 we have that∑j(j− l+Ei∆t/∆v)Np(j− l+Ei∆t/∆v) = Mp the first moment of the cardinal spline

of degree p. Then ∑j

jfk+1i,j (vj) =

∑l

cl(Mp + l − Ei∆t/∆v).

On the other hand the total momentum on the column at the beginning of the time stepscan also be expressed with the same spline coefficients simply using the same calculationwith Ei = 0. Thus ∑

j

jfki,j(vj) =∑l

cl(Mp + l).


It follows that ∑j

vjfk+1i,j (vj) =

∑j

vjfki,j(vj)−∆t

∑l

clEi.

From the proof of the previous proposition we recall that ∆v∑

l cl = ∆v∑

j fki,j = ni.

So that we have conservation of total momentum on this split step provided∑i

Eini = 0

which concludes the prove the the proposition.

As E is linked to n via the Poisson solver this property needs to follow from thenumerical scheme for the Poisson equation. Note that

∑iEini = 0 is equivalent to∑

iEiρi = 0 on a periodic domain provided∑

iEi = 0 which should be the case becauseE is a gradient. This is a discrete version of

∫E(x)ρ(x) dx which we have shown to

vanish on a periodic domain.

Proposition 11. The FFT spectral Poisson solver introduced previously satisfies∑iEi = 0 and

∑iEiρi = 0.

Proof. First by definition of the discrete Fourier Transform∑

iEi =√NE0 which

is set to 0 in the algorithm. Then using the discrete Parseval inequality, noticing thatρi and Ei are real while there Fourier transforms are not, we have∑

j

ρjEj =

N/2−1∑k=−N/2

ρk¯Ek =

N/2−1∑k=−N/2

ik|Ek|2 =

N/2−1∑k=1

ik(|Ek|2 − |E−k|2),

as the algorithm yields ikEk = ρk and E−N/2 = 0 by construction. On the other hand

Ek =1√N

N−1∑j=0

Eje2iπjk/N

from which it easily follows as the Ej are real that E−k =¯Ek and thus they have the

same modulus.

3.3. A semi-Lagrangian method without splitting. Time split semi-Lagrangianmethods for the Vlasov-Poisson equations have the great advantage of boiling down ateach split step to constant coefficient advections, which enable an exact computationof the origin of the characteristics and thus greatly simplifies the algorithm. Howeverthe splitting itself is a source of errors giving a even greater importance to the axesdirections. In some cases it is interesting to develop a semi-Lagrangian method withoutsplitting. In this case the origins of the characteristics need to be computed numeri-cally as the solutions for different initial conditions of the following ordinary differentialequation (ODE):

dV

dt= E(X(t), t),

dX

dt= V.

Note that this ODE needs to be solved backward in time as we are backtracking thecharacteristics. The algorithm enabling to go from time step tn to time step tn+1 is thenthe following:at time tn we know fn and En at the grid points and we want to compute the samevalues at time tn+1. An order 2 predictor-corrector scheme can be defined as follows:

(1) Predict a value En+1 for the electric field at time tn+1.(2) For all grid points xi = Xn+1, vj = V n+1 compute successively


• V n+1/2 = V n+1 − ∆t2 E

n+1(Xn+1),

• Xn = Xn+1 −∆tV n+1/2,• V n = V n+1/2 − ∆t

2 En(Xn).

• Interpolate fn at point (Xn, V n).(3) We then have a first approximation of fn+1(xi, vj) = fn(Xn, V n) that can be

used to compute a corrected version of En+1 from which a new iteration canbe performed if necessary to improve the precision.

In order to initialize the prediction of En+1, it is convenient to use the continuity equationthat is obtained by integrating the Vlasov equation with respect to the velocity variable:

∂ρ

∂t+∇ · J = 0.

with an order 2 centred scheme, we then obtain an approximation of ρn+1 from ρn−1 =∫fn−1 dv and of Jn =

∫vfn dv in the form

ρn+1 = ρn−1 − 2∆t∇ · Jn.We then compute En+1 solving the Poisson equation with the source term ρn+1.

More generally for other Vlasov type equations like the guiding-center equation, thedrift-kinetic or the gyrokinetic equations, it is possible to proceed in the same mannerusing the first velocity moments of the Vlasov equation to predict the electric field attime tn+1.

2D spline interpolation.

3.4. Importance of conservativity. Consider an abstract Vlasov equation in theform

∂f

∂t+ A(z, t) · ∇zf = 0,

with ∇ · A = 0, where z represents here all the phase space variables. As we haveseen, the property ∇ ·A = 0 implies the conservativity of the equation. Consider nowa splitting obtained by decompossing z into two groups of variables z1 and z2, we shallcall the corresponding components of A, A1 and A2. We then need to solve successively

∂f

∂t+ A1(z, t) · ∇x1f = 0,

and∂f

∂t+ A2(z, t) · ∇x2f = 0.

We have ∇ ·A = ∇z1 ·A1 +∇z2 ·A2 = 0, but in general ∇z1 ·A1 and ∇z2 ·A2 are notboth vanishing in which case none of the two split equations is conservative. It will thenbe very challenging to derive a conservative numerical method for the split system.

Examples.

(1) The Vlasov-Poisson model. In this case A = (v,E(x, t)). Splitting in theclassical manner between x and v, we obtain A1 = v and A1 = E(x, t). wehave in this case∇x·A1 = 0 and∇v·A2 = 0, so that the splitting is conservative.

(2) The guiding center model. This is a classical model in magnetized plasmaphysics which reads

∂ρ

∂t+ vD · ∇ρ = 0, −∆φ = ρ,

with

vD =−∇φ×B

B2=

(−∂φ∂y∂φ∂x

)if B = ez unit vector in the z-direction.


We have indeed ∇ · vD = 0, so that the guiding center model is conservative.However, splitting in the x and y directions, we obtain

∂ρ

∂t− ∂φ

∂y

∂ρ

∂x= 0,

and∂ρ

∂t+∂φ

∂x

∂ρ

∂x= 0.

The cross derivative ∂2φ∂x∂y does in general not vanish and therefore the splitting

is not conservative in this case. When numerical simulations are performedusing non conservative split equations, large variations of total particle density(which should be conserved) can be observed, in particle in regions of the sim-ulation where the distribution function is not well resolved. This phenomenonwill happen in most problems modelled by the Vlasov equations, are there arefilaments appearing and then vortex roll-ups. An illustration of this phenom-enon is displayed in Figure 3 where the evolution of the L1 and L2 norms iscompared for methods with and without splitting and also for a conservativemethod with splitting that shall be introduced in the next section. We observethat the methods without splitting and the conservative split method show agood physical behaviour, whereas for the split non-conservative method, the to-tal number of particles varies by a very large amount which renders the resultscompletely unphysical and this method unacceptable.

3.5. The conservative semi-Lagrangian method. It is also possible to derivesemi-Lagrangian methods using the conservative form of the Vlasov equation. Thesewill then naturally be conservative.

Let us point out that the classical semi-Lagrangian method applied to the Vlasovequation is also exactly conservative. This can be proved by showing that the resultingscheme is algebraically equivalent to a conservative scheme. See [19] for details.

The conservative semi-Lagrangian method has similarities with a Finite Volumemethod, but the computation of the fluxes is replaced by an integration over the volumeoccupied at the previous time step tn by the cell under consideration. The unknown isthe average value of f in one cell 1

|V |∫V f dx dv and, as for finite volumes the numerical

algorithm consists of three stages:

(1) Reconstruction of a polynomial approximation of the desired degree from thecell averages.

(2) Backtrack the cell down the flow of the characteristics (generally only the cornerpoints are backtracked and the origin cell is approximated by a quadrilateral).

(3) Compute the cell average of f at tn+1 using that∫V f dx dv is conserved along

characteristics.

A scheme of principle is given in figure 4.As in this case we work on the conservative form of the equation, the split equations

are also in conservative form and thus the splitting does not generate conservativityissues, and as has been shown, the split conservative method performs well, unlike thesplit method based on the advective form, see figure 3.

For this reason, and also because the conservative method becomes very simple inthis case, as the 1D cell is completely determined by its enpoints, we shall only use itfor split 1D equations in the conservative form

∂f

∂t+

∂

∂x(a(x, t)f) = 0.


-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 200 400 600 800 1000

’diagt1kh2.plot’ u 1:3’diagt16kh2.plot’ u 1:3’diagt17kh2.plot’ u 1:3

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 200 400 600 800 1000

’diagt1kh2.plot’ u 1:4’diagt16kh2.plot’ u 1:4’diagt17kh2.plot’ u 1:4

Figure 3. Evolution of a Kelvin-Helmhotz instability for the guiding-center model. The top figure displays a snapshot of the distributionfunction during the creation of a vortex. The snapshot is taken a thetime corresponding to the large increase of the L1 norm on the bottomleft figure. The bottom figures represent the evolution in time of the L1

(left) and L2 (right) norms for the non conservative splitting (top curve),conservative splitting (middle curve) and without splitting (bottomcurve).

Figure 4. Idea of conservative semi-Lagrangian method.


Let us now detail the 3 steps of the algorithm in the 1D case starting with step 1.This step consists in construction on each cell a polynomial of degree m which has agiven average value. The classical technique to do this consists in reconstructing theprimitive of the polynomial we are looking for as follows:Let fnj be the fixed average value of fn in the cell [xj− 1

2, xj+ 1

2] of length hj = xj+ 1

2−xj− 1

2.

We wish to construct a polynomial pm(x) of degree m such that

1

hj

∫ xj+ 1

2

xj− 1

2

pm(x) dx = fnj .

In order to do this we shall define pm(x) verifying ddx pm(x) = pm(x) so that

hjfnj =

∫ xj+ 1

2

xj− 1

2

pm(x) dx = pm(xj+ 12)− pm(xj− 1

2).

Let W (x) =∫ xx 1

2

fn(x) dx be a primitive of the piecewise constant function fn which

takes the value fnj on [xj− 12, xj+ 1

2]. We then have

W (xj+ 12) =

j∑k=1

hkfnk

and

W (xj+ 12)−W (xj− 1

2) = hjf

nj = pm(xj+ 1

2)− pm(xj− 1

2).

Let us take for pm an interpolation polynomial at points xj+ 12

of the function W , which

will yield that

1

hj

∫ xj+ 1

2

x 12

pm(x) dx =1

hj(pm(xj+ 1

2)− pm(xj− 1

2))

=1

hj(W (xj+ 1

2)−W (xj− 1

2))

= fnj ,

which is what we wanted.It remains to choose the type of interpolation. The simplest way is to use Lagrange

interpolation with as many neighboring points as needed for the chosen degree. Onecan then choose a centered stencil or try an choose a stencil enabling to reduce theoscillations. ENO type stencil have not proved efficient for the Vlasov equation, as theyincrease the diffusivity, but well designed WENO methods have had some success [37,27, 28]. Note that, the Vlasov equation generates subcell oscillations but no shocks, theissue for designing good limiters are therefore different than in traditional conservationlaws arising in fluid dynamics. Limiters are also needed for enforcing positivity, in thePFC algorithm [21] only such very weak limiting is performed.

It is also possible to chose a global interpolation of spline type which will enableto have more regularity on the reconstruction. Indeed for a Lagrange interpolation,the primitive will be continuous and the so the reconstructed polynomial pm will bediscontinuous at cell boundaries. On the other hand, if the primitive is for examplea cubic spline, it will be on each cell a polynomial of degree 3 and be globally of C2

regularity. The the reconstructed polynomial pm will be a quadratic spline, which is apolynomial of degree 2 within each cell and of global regularity C1.


The second step of the method consists in computing the origin of the characteristicsending at the grid points. This step is in principle identical as for the classical semi-Lagrangian algorithm. For constant coefficient advections an exact solution can becomputed. Note that for non constant (in the x variable) coefficient advection, the

point based semi-Lagrangian algorithm using the advective form of the equation ∂f∂t +

a((t, x)∂f∂t = 0 is not conservative and not equivalent to the conservative form ∂f∂t +

∂∂t(a(t, x)f) = 0.

A difficulty for the numerical method is that we are backtracking the characteristicsand that in our Vlasov type problems the advection field a depends non linearly on f ,at least through the electric field, which is generally not known, but can be predicted attime tn+1. A robust and simple second order algorithm for computing the origin of thecharacteristics is the following. Assuming a is a known function of t and x. We get asecond order approximation of the solution X(tn) = x∗i at time tn of dX

dt = a(t,X) withX(tn+1) = xi using the trapezoidal rule (a midpoint rule would also work and give thesame order):

(51) xi − x∗i =∆t

2[a(tn+1, xi) + a(tn, x

∗i )].

This is in general an implicit equation for x∗i . However, in our applications a is knownonly at grid points, so an interpolation procedure is necessary to compute a(tn, x

∗i ) as x∗i

is in general not a grid point. Moreover, in most cases that we have been investigating,no gain is obtained by using more that linear interpolation. And in this case, as describedin [18] a completely explicit formula can be obtained: If we denote by xi0 the for now

unknown grid point immediately to the left of x∗i and by βi =x∗i−xi0

∆x , we have βi ∈ [0, 1[.Then we also have xi−x∗i = (i− i0−β)∆x, so that if we can determine i0, and βi, we getx∗i . Now if we inject this relation, and approximate a(tn, x

∗i ) by a linear interpolation in

the cell xi0 , xi0+1, we get

i− i0 − βi =∆t

2∆x[a(tn+1, xi) + (1− βi)a(tn, xi0) + βia(tn, xi0+1)].

From this we can extract the following formula for βi:

βi =i− i0 − ∆t

2∆x [a(tn+1, xi) + a(tn, xi0)]

1 + ∆t2∆x [a(tn, xi0+1)− a(tn, xi0)]

.

This formula is valid as long as the denominator does not vanish. This brings us tothe question of stability of the semi-Lagrangian algorithm. It relies on the fact thatthe Lagrangian grid obtained by backtracking all the original grid points along thecharacteristics remains an acceptable grid. In 1D, this boils down to saying that theorder of the grid points needs to be preserved and that those should not get to closeto each other. We express this condition by x∗i+1 − x∗i > tol, where tol is some smallpositive tolerance.

The third and last step consists in computing the average value on the cell at timetn+1 by using the relation∫ x

i+ 12

xi− 1

2

fn+1(x) dx =

∫ X(tn;xi+ 1

2,tn+1)

X(tn;xi− 1

2,tn+1)

fn(x) dx,

where fn(x) is the polynomial function reconstructed on each cell in step 1.

4. PARTICLE METHODS 47

Positivity. Conservative semi-Lagrangian can benefit during the reconstruction stepof a filtering procedure in the same way this is done in traditional finite volume method,preserving the conservativity of the the method. Note however that most filters usedfor fluid dynamics problems are too strong and two dissipative for Vlasov type problemswhere no shocks occur. For many problems it is enough to use a filter just to ensurethe positivity of the distribution function. A filter enabling to conserve positivity isdescribed in [21, 18].

4. Particle methods

4.1. Introduction. The method which is still by far the most used method for thesimulation of the Vlasov-Maxwell equations is the Particle In Cell (PIC) method whichconsists in the coupling of a particle method for the Vlasov equation and a mesh basedmethod for the computation of the self-consistent field using Maxwell’s equations or somereduced model. The principle of the method is to discretize the distribution functionby a collection of macro-particles representing the intial distribution function f0(x,v)which, when normalized such that its integral is 1, represents a probability density. Themacro-particles are then advanced in time by solving the equations of motion of theparticles in the global electromagnetic field. Coupling the field solver with the particlesis done by computing the sources of Maxwell’s equations ρ and J from the particles usingsome regularization method. Any classical solver for Maxwell’s equations can then beused on the mesh. In order to continue the time loop the fields need then to be computedat the particle positions, which can be done in a natural way using some solvers (FiniteElements for example), where the discrete fields are defined at any place. In order casesome interpolation procedure needs to be defined. A huge literature on these methodsexists, including two books that are rather physics oriented, by Birdsall and Langdon [8]and Hockney and Eastwood [24]. Mathematical convergence proofs of the algorithmshave also been performed in some special cases, see Neunzert and Wick [26], Cottet andRaviart [17], Victory and Allen [36] and Wollman [42].

There also exists a variant of the PIC method which is often used when the physicsthat is being investigated remains close to some equilibrium configuration, examples arePIC simulations of tokamak plasmas or of particle accelerators. This method is calledδf . It consists in expanding the distribution function in the neighborhood of a knownequilibrium f0 in f = f0 + δf and to approximate only the δf part with a PIC method.Another particle method, linked to SPH (smooth particle hydrodynamics) used in fluiddynamics has been introduced by Bateson and Hewett [4], but seems not to have beenused very much since. It consists in pushing a relatively small number of macro-particlesin the form of a Gaussian whose size can vary and that interact directly with each other.

4.2. The PIC method. The principle of a particle method is to approximate thedistribution function f solution of the Vlasov equation by a sum of Dirac masses centeredat the particle positions in phase space (xk(t),vk(t))1≤k≤N of a number N of macro-particles each having a weight wk. The approximated distribution function that wedenote by fN then writes

fN (x,v, t) =

N∑k=1

wkδ(x− xk(t)) δ(v − vk(t)).

Positions x0k, velocities v0

k and weights wk are initialized such that fN (x,v, 0) is anapproximation, in some sense that remains to be precized, of the intial distributionfunction f0(x,v). The time evolution of the approximation is done by advancing the


macro-particles along the characteristics of the Vlasov equation, i.e. by solving thesystem of differential equations

dxkdt

= vk

dvkdt

=q

m(E(xk, t) + vk ×B(xk, t))

xk(0) = x0k, vk(0) = v0

k.

Proposition 12. The function fN is a solution in the sense of distributions of theVlasov equation associated to the initial condition f0

N (x,v) =∑N

k=1wkδ(x − x0k) δ(v −

v0k).

Proof. Let ϕ ∈ C∞c (R3 × R3×]0,+∞[). Then fN defines a distribution of R3 ×R3×]0,+∞[ in the following way:

〈fN , ϕ〉 =N∑k=1

∫ T

0wkϕ(xk(t),vk(t), t) dt.

We then have

〈∂fN∂t

, ϕ〉 = −〈fN ,∂ϕ

∂t〉 = −

N∑k=1

wk

∫ T

0

∂ϕ

∂t(xk(t),vk(t), t) dt,

butd

dt(ϕ(xk(t),vk(t), t)) =

dxkdt· ∇xϕ+

dvkdt· ∇vϕ+

∂ϕ

∂t(xk(t),vk(t), t),

and as ϕ has compact support in R3 × R3×]0,+∞[, it vanishes for t = 0 and t = T . So∫ T

0

d

dt(ϕ(xk(t),vk(t), t)) dt = 0.

It follows that

〈∂fN∂t

, ϕ〉 =

N∑k=1

wk

∫ T

0(vk · ∇xϕ+

q

m(E(xk, t) + vk ×B(xk, t)) · ∇vϕ) dt

= −〈v · ∇xfN +q

m(E(xk, t) + v ×B(xk, t)) · ∇vfN , ϕ〉.

Which means that fN verifies exactly the Vlasov equation in the sense of distributions.

Consequence: If it is possible to solve exactly the equations of motion, which issometimes the case for a sufficiently simple applied field, the particle method gives theexact solution for an initial distribution function which is a sum of Dirac masses.

The self-consistent electromagnetic field is computed on a mesh of physical spaceusing a classical method (e.g. Finite Elements, Finite Differences, ...) to solve theMaxwell or the Poisson equations.

In order to determine completely a particle method, it is necessary to precise howthe initial condition f0

N is chosen and what is numerical method chosen for the solutionof the characteristics equations and also to define the particle-mesh interaction.

Let us detail the main steps of the PIC algorithm:


Choice of the initial condition.

• Deterministic method: Define a phase space mesh (uniform or not) and pickas the initial position of the particles (x0

k,v0k) the barycentres of the cells and

for weights wk associated to the integral of f0 on the corresponding cell: wk =∫Vkf0(x,v) dxdv so that

∑k wk =

∫f0(x,v) dxdv.

• Monte-Carlo method: Pick the intial positions in a random or pseudo-randomway using the probability density associated to f0.

Remark 12. Note that randomization occurs through the non-linear processes, whichare generally such that holes appear in the phase space distribution of particles when theyare started from a grid. Moreover the alignment of the particles on a uniform grid canalso trigger some small physical, e.g. two stream, instabilities. For this reason a pseudo-random initialization is usually the best choice and is mostly used in practice.

Particle-Mesh coupling. The particle approximation fN of the distribution functiondoes not naturally give an expression for this function at all points of phase space. Thusfor the coupling with the field solver which is defined on the mesh a regularizing stepis necessary. To this aim we need to define convolution kernels which can be used thethis regularization procedure. On cartesian meshes B-splines are mostly used as thisconvolution kernel. B-splines can be defined recursively: The degree 0 B-spline that weshall denote by S0 is defined by

S0(x) =

1

∆x si − ∆x2 ≤ x <

∆x2 ,

0 else.

Higher order B-splines are then defined by:For all m ∈ N∗,

Sm(x) = (S0)∗m(x),

= S0 ∗ Sm−1(x),

=1

∆x

∫ x+ ∆x2

x−∆x2

Sm−1(u) du.

In particular the degree 1 spline is

S1(x) =

1

∆x(1− |x|∆x) si |x| < ∆x,0 sinon,

the degree 2 spline is

S2(x) =1

∆x

12(3

2 −|x|∆x)2 si 1

2∆x < |x| < 32∆x,

34 − ( x

∆x)2 si |x| < 12∆x,

0 sinon,

the degree 3 spline is

S3(x) =1

6∆x

(2− |x|∆x)3 si ∆x ≤ |x| < 2∆x,

4− 6(x

∆x

)2+ 3

(|x|∆x

)3si 0 ≤ |x| < ∆x,

0 sinon.

B-splines verify the following important properties

Proposition 13. • Unit mean∫Sm(x) dx = 1.


• Partition of unit. For xj = j∆x,

∆x∑j

Sm(x− xj) = 1.

• Parity

Sm(−x) = Sm(x).

The sources for Maxwell’s equations ρ and J are defined from the numerical distri-bution function fN , for a particle species of charge q by

ρN = q∑k

wkδ(x− xk), JN = q∑k

wkδ(x− xk)vk.

We then apply the convolution kernel S to defined ρ and J at any point of space and inparticular at the grid points:

ρh(x, t) =

∫S(x− x′)ρN (x′) dx′ = q

∑k

wkS(x− xk),(52)

Jh(x, t) =

∫S(x− x′)JN (x′) dx′ = q

∑k

wkS(x− xk)vk.(53)

In order to get conservation of total momentum, when a regularization kernel isapplied to the particles, the same kernel needs to be applied to the field seen as Diracmasses at the grid points in order to compute the field at the particle positions. Wethen obtain

(54) Eh(x, t) =∑j

Ej(t)S(x− xj), Bh(x, t) =∑j

Bj(t)S(x− xj).

Note that in the classical case where S = S1 this regularization is equivalent to a linearinterpolation of the fields defined at the grid points to the positions of the particles, butfor higher order splines this is not an interpolation anymore and the regularized field atthe grid points is not equal to its original value Ej anymore, but for example in the caseof S3, to 1

6Ej−1 + 23Ej + 1

6Ej+1.Conservation properties at the semi-discrete level.

• Conservation of mass. The discrete mass is defined as∫fN (x,v, t) dxdv =∑

k wk. This is obviously conserved if no particle gets in or out of the domain,as wk is conserved for each particle when the particles move.• Conservation of momentum. The total momentum of the system is defined as

P = m

∫vfN (x,v, t) dxdv =

∑k

mkwkvk(t).

SodPdt

=∑k

mkwkdvkdt

=∑k

wkqkEh(xk, t).

In the case Eh is computed using a Finite Difference approximation, its valueat the particle position should be computed using the same convolution kernelas is used for computing the charge and current densities from the particlepositions. Then Eh(xk, t) =

∑j Ej(t)S(xk − xj) and so

dPdt

=∑k

wkqk∑j

Ej(t)S(xk − xj).


Then exchanging the sum on the grid points i and the sum on the particles kwe get

dPdt

=∑j

Ej(t)∑k

wkqkS(xk − xj) =∑j

Ej(t)ρj(t),

so that the total momentum is conserved provided the field solver is such that∑j Ej(t)ρj(t).

In the case of a Finite Element PIC solver the Finite Element interpolantnaturally provides and expression of the fields everywhere in the computationaldomain and the weak form of the right-hand side provides a natural definitionof the source term for the finite element formulation. Let us also check theconservation of momentum in this case. Denoting by ϕi the Finite Elementbasis functions, we have Eh(xk, t) =

∑j Ej(t)ϕj(xk) and so

dPdt

=∑k

wkqk∑j

Ej(t)ϕj(xk) =∑j

Ej(t)ρj(t)

where ρj =∫qfN (xk,vk, t)ϕj(x) dxdv.

Remark 13. Note the conservation of momentum is linked to the self-force problemthat is often mentioned in the PIC literature. Indeed if the system is reduced to oneparticle. The conservation of momentum is equivalent to the fact that a particle does notapply a force on itself.

Time scheme for the particles. Let us consider first only the case when the magneticfield vanishes (Vlasov-Poisson). Then the macro-particles obey the following equationsof motion:

dxkdt

= vk,dvkdt

=q

mE(xk, t).

This system being hamiltonian, it should be solved using a symplectic time scheme inorder to enjoy long time conservation properties. The scheme which is used most of thetime is the Verlet scheme, which is defined as follows. We assume xnk , vnk and Enk known.

vn+ 1

2k = vnk +

q∆t

2mEnk(xnk),(55)

xn+1k = xnk + ∆tv

n+ 12

k ,(56)

vn+1k = v

n+ 12

k +q∆t

2mEn+1k (xn+1

k ).(57)

We notice that step (57) needs the electric field at time tn+1. It can be computed afterstep (56) by solving the Poisson equation which uses as input ρn+1

h that needs only xn+1k

and not vn+1k .

Time loop. Let us now summarize the main stages to go from time tn to time tn+1:

(1) We compute the charge density ρh and current density Jh on the grid usingrelations (52)-(53).

(2) We update the electromagnetic field using a classical mesh based solver (finitedifferences, finite elements, spectral, ....).

(3) We compute the fields at the particle positions using relations (54).(4) Particles are advanced using a numerical scheme for the characteristics for ex-

ample Verlet (55)-(57).


Probabilistic interpretation of the PIC method. Even when the particles are initial-ized in a non random manner, the non linear interactions along with numerical errorsintroduce random effects after some time. The right tool for the mathematical inves-tigation of PIC methods is thus probability and the analysis is similar to this of otherMonte-Carlo numerical methods.

A Monte-Carlo description of the PIC method can be introduced as follows: We startby drawing macro-particles in phase space as a random realization of the probability lawassociated to the probability density p obtained from the intial distribution f0 by

p =1

Nf0, where N =

∫f0 dx dv.

When the particles are initialized, they are advance using the equations of motion whichare deterministic, but at each time t the particles given by their positions in phasespace (xk(t),vk(t))1≤k≤N represent a realization of the probability law associated to thedensity p(x,v, t) = 1

N f(x,v, t) linked to the solution at time t of the Vlasov equation.In this framework, the different physical variables can be expressed as expectancies

under the probability law of density p. For a given function g its expectancy is thendefined by

Ep(g) =

∫g(x,v)p(x,v, t) dx dv =

1

N

∫g(x,v)f(x,v, t) dx dv.

Thanks to the law of large numbers, these expectancies can be approximated by randomrealizations:

Ef (g(x,v)) ≈ 1

N

N∑k=1

g(xk,vk).

Moreover, the central limit theorem enables to obtain an error estimate Ef (g(x,v)2) <+∞, and so the error

εN = Ef (g(x,v))− 1

N

N∑k=1

g(xk,vk)

is such that√Nσ εN converges to the centered reduced Gaussian whose variance σ is given

by

σ2 = Ef (g(x,v)2)− Ef (g(x,v))2,

=

∫g(x,v)2f(x,v) dx dv −

(∫g(x,v)f(x,v) dx dv

)2.

We can deduce that this approximation converges as 1/√N when N goes to +∞ and

the approximation is all the better that the variance is small. In particular variancereduction techniques used in statistics are a mean to improve the approximation.

The different physical quantities can be interpreted thanks to the probabilistic ter-minology. The kinetic energy is

NEf (|v|2) =

∫|v|2f(x,v) dx dv,

the charge density at point xi is

ρi = NEf (Si) =

∫S(x− xi)f(x,v) dx dv,

and the current density at point xi is

Ji = NEf (vSi) =

∫S(x− xi)vf(x,v) dx dv.


Moreover these expectancies can be approximated with a random realization in thefollowing way:

Ef (|v|2) ≈ 1

N

∑v2k, ρi = Ef (Si) ≈

1

N

∑S(xk − xi),

Ji = Ef (vSi) ≈1

N

∑S(xk − xi)vk.

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eric sonnendruc ker max-planck-institut fur plasmaphysik

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