an optimal vlasov-fokker-planck solver for simulation of ... · outline • motivations ......

U N C L A S S I F I E D

An optimal Vlasov-Fokker-Planck solver for simulation of kinetic ICF plasmas

L. Chacón1, W.T. Taitano2, A.N. Simakov2

1T-5, Applied mathematics and plasma physics group 2XCP-6, plasma theory and applications group

Thermonuclear Burn Initiative

Kinetic Effects in ICF Workshop Livermore, CA, April 5-7 2016


Outline

•  Motivations and challenges of fully kinetic ion simulations

•  Enabling Technologies –  Fully implicit time stepping + scalable solver (MG precond

JFNK) –  Exact discrete conservation properties –  Phase-space adaptivity: vth adaptivity + Lagrangian mesh

•  Progress report: verification, spherical geometry (shocks), kinetic effects at interfaces


Kinetic effects may be important in ICF

Hydro Kinetic

YO

C⌘

Yie

ldexperim

ental

Yie

ldhydro,no

mix

Figure from M.J. Rosenberg, PRL (2014)


We consider fully kinetic ions + fluid electrons

Vlasov-Fokker-Planck

r2vHj (~v) = �8⇡fj (~v)

r2vGj (~v) = Hj (~v)

Dj = rvrvGj Aj = rvHj

Cij (fi, fj) = �ijrv · [ Dj ·rvfi �mi

mjAjfi ]

DfiDt

⌘ @fi@t

+ ~v ·rfi + ~ai ·rvfi =X

j

Cij (fi, fj)

Fluid electrons

3

2@t

(ne

Te

) +5

2@x

(ue

ne

Te

)� ue

@x

(ne

Te

)� @x

e

@x

Te

=X

↵

Ce↵

ue = �q�1e n�1

e

NsX

↵ 6=e

q↵n↵u↵ne = �q�1e

NsXq↵n↵

Electric field model: e pressure, friction, thermal forces

E = �rpe +

Pi Fie

ene= �rpe

ene� ↵0(Zeff )me

e

X

i

⌫ei(Ve �Vi)��0(Zeff )

erTe


ICF kinetic simulation tools are sparse

•  French CEA’s FPion [1] and FUSE [2] codes: •  Semi-implicit (Δt ~ τcol e.g. can’t study pusher mix) •  Adaptive grid (r-v), but non-conservative, single-species velocity

•  Recent implosion calculation using the LSP code [3] •  Hybrid PIC code •  Difficulties with spherical geometry •  Nanbu collision operator (suspect when using Δt>τcol [4])

•  Our approach: •  Fully nonlinearly implicit (Δt >> τcol) •  Multispecies adaptive grid (r-v) using local thermal velocity/species •  Fully conservative (mass, momentum, energy)

[1] O. Larroche, EPJ 27, pp.131-146 (2003) [2] B. Peigney et al., JCP 278 (2014) [3] T.J.T. Kwan et al., IFSA2015, Seattle WA (2015) [4] A.M. Dimits et al., JCP 228, pp.4881-4892 (2009)


Challenges of fully kinetic simulations for ICF

v

x

Under-resolved

cold distribution

Resolved hot

distribution

vth,hot

vth,cold

•  Disparate temperatures during implosion dictate velocity resolution. –  vth,max determines Lv

–  vth,min determines Δv

•  Shock width and capsule size dictate physical space resolution


Numerical resolution challenges

•  Phase space resolution challenges –  Intra species vth,max /vth,min~100 –  Inter species (vth,α /vth,β)max~30 –  Nv~ [10(vth,max/vth,min)x(vth,α /vth,β)]2 ~109 –  Nr ~ 103-104 –  N=NrNv~1012-1013 unknowns

•  Temporal resolution challenges –  tsim=1 ns –  Nt=109 time steps

3 Numerical Challenges and Algorithms 3

becomes quite restrictive. In a typical shell implosion, the radius decreases from a mm to a few tensof µm, about 10-30 times. Assuming the lower limit, a static mesh would have to provide su�cientresolution at the final point of compression, say 50-100 points. This, in turn, would require aninitial uniform static radial mesh of:

Nr

⇠ 103.

Temporal resolution. Challenges in the temporal integration of VFP stem from the two mainphenomena at play: streaming and collisions. These have to be compared against the total simu-lation time, i.e., the implosion time. Assuming an initial radius R of 1mm, and a typical implosionvelocity v

i

of 20 cm/µs, we find:

⌧i

⇡ R

vi

⇠ 5 ns.

The temporal stability limit for integrating collisions with an explicit time integration scheme isgiven by the corresponding Courant condition:

�tcoll

exp

⇠ �v2

D⇠ �v2

v2

th

⌫coll

.

We take typical conditions at the end of the compression phase for density and temperature,

ni

⇡ 200 g/cm3 ⇡ 5⇥ 1025 cm�3 , Ti

⇡ 2.5 keV.

For these conditions, we have vth

⇡ 10vmin

th

, and the ion collision frequency is:

⌫coll

= 4.8⇥ 10�8

✓

mp

mi

◆

1/2 ni

(cc�1) ln⇤T

i

(eV )3/2

s�1 ⇡ 105 ns�1,

where we have assumed ln ⇤ ⇠ 10. It follows that:

�tcoll

exp

⇠ 110

✓

�v

vmin

th

◆

2

⌫�1

coll

⇠ 10�9 ns,

where we have used Eq. 1. Time steps for the early compression phase will be orders of magnitudemore restrictive even though the plasma is less dense (n

i

⇠ 0.225 g/cm3), because it is significantlycolder (T

i

⇠ 1 eV). This result implies that an explicit scheme, for the conditions and velocity-spaceresolution stated, will require > 109 � 1010 time steps to reach the implosion time scale.

Regarding streaming, the time step explicit stability limits of advection in physical and velocityspace are comparable, and are given by the corresponding CFL constraint for the maximum velocityunder consideration in our discretization, v

max

⇠ 10vmax

th

:

�tstrexp

⇠ �r

vmax

⇠ R

Nr

10vmax

th

.

For Tmax

⇡ 100 keV , we have vmax

th

⇠ 3 ⇥ 108 cm/s. Therefore, for R ⇡ 0.1 cm and Nr

⇠ 103, wefind:

�tstrexp

⇠ 10�4 � 10�5 ns.

Thus, we find streaming to be significantly less restrictive numerically than collisions.


•  Mesh adaptivity: –  v-space adaptivity with vth normalization, Nv~104-105 –  Lagrangian mesh in physical space, Nr~102

–  N=NvNr~106~107

•  Implicit time-stepping: –  JFNK nonlinear solver –  Multigrid preconditioning [1,2] –  Δtimp=Δtstr~10-3 ns, Nt~103-104

•  Exact conservation properties (nonlinear discretization)

Our approach: Implicit time-stepping and adaptive meshing, exact conservation properties

[1] Chacon et al., JCP 157, pp. 618-653 (2000) [2] Chacon el al., JCP 157, pp. 654-682 (2000)

v = v/vth,↵


Implicit time-stepping for collision operator: nonlinear solver and preconditioning

•  JFNK as nonlinear solver [3]:

•  Right preconditioning strategy: •  Block diagonalization with lagged coefficients:

fk = fk�1 � JkRk�1

Jk�P k�1

��1P k�1�fk = �Rk�1

[3] D.A. Knoll et al., JCP 193, pp. 357-397 (2004)

R = @tf↵ �NsX

�

r ·hD� ·rvf↵ � ~A�f↵

i

P k�1 = @t � �NsX

�

r ·hD� ·rv � � ~A��

i


MG preconditioner keeps linear iterations bounded


Implicit solver performance is O(Nv)!

Nv104 105

CPU

[sec

]

104

106

108

ImplicitprecO(Nv)ExplicitO(N2

v)

4 orders of magnitude more efficient than explicit methods

[4] W.T. Taitano, JCP 297, pp.357-380 (2015)

Solver CPU time versus size of unknown


Exact energy conservation in collision operator is critical for accuracy

•  For demonstration, we consider electron-proton thermalization

•  More details in Taitano’s talk


Adaptivity in velocity space: vth adaptivity

v

x

Resolved cold

distribution

Resolved hot

distribution

dvth/dt > 0

dvth/dt < 0

v

x

Under-resolved

cold distribution

Resolved hot

distribution

vth,hot

vth,cold

Static Mesh vth adaptive Mesh


Adaptivity in physical space: Lagrangian mesh evolves with capsule


Phase-space mesh adaptivity: exact transformation of FP equation

1D spherical (with logical mesh); 2D cylindrical geometry in velocity space 624 CHACON ET AL.

FIG. 1. Diagram of the local cylindrical velocity coordinate system (vr , vp) considered in this work. Cylin-drical symmetry is assumed. The spherical radius vector r is included for reference.

by-product, the required numerical representation of the subordinate problems for H andG, which will be somewhat different from the traditional treatment.

3.1. Definition of the Computational Velocity Domain

So far, the discussionhas been independent of a particular geometry and/or dimensionalityin velocity space. To focus the discussion that follows, a 2D cylindrical velocity space withangular symmetry is adopted. This space is spanned by (vr , vp), where vr is the cylindricalz-axis, and vp is the cylindrical r -axis (Fig. 1), and vr ∈ [0, vlimit]; vp ∈ [0, vlimit]. Here, vlimitis typically set to several times the characteristics velocity of the problem, v0.The domain is discretized with an integer mesh and a half mesh (Fig. 2). The integer

mesh is defined using Nr (+1) nodes in the vr axis, and Np(+1) nodes in the vp axis, withthe constraints

vr,1 = 0, vr,Nr = vlimit

vp,1 = 0, vp,Np = vlimit.

Each velocity node is characterized by a pair (vr,i , vp, j ), with i = 1, . . . , Nr (+1), andj = 1, . . . , Np(+1). The additional (i = Nr + 1, j) and (i, j = Np + 1) nodes at the bound-aries will serve a double purpose: (1) they will be used to impose the far-field boundaryconditions for the Rosenbluth potentials, and (2) they will allow an accurate determina-tion of the friction and diffusion coefficients of the Fokker–Planck collision operator at the

FIG. 2. Diagram of the 9-point stencil in velocity space employed in the discretization of the Fokker–Planckcollision operator.

Coordinate transformation:

Jacobian of transformation:

Always fixed

Ẋ = dr/dt (mesh speed)

rt,0

rt,1

ξ

bv|| ⌘~v ·~brvth,↵

, bv? ⌘

qv2 � v2||

vth,↵

pgv (t, r, bv?) ⌘ v3th,↵ (t, r) r2bv?

~v||

~v?


Fokker-Planck equation: inertial terms

•  VRFP equation in transformed coordinates

Inertial terms due to vth adaptivity and Lagrangian mesh

bv|| = �

bv||2

⇣v

�2th,↵@tv

2th,↵ + J

�1r⇠

⇣bv|| � b

x

⌘v

�1th,↵@⇠v

2th,↵

⌘+

bv2?vth,↵r

+q↵E||

Jr⇠m↵vth,↵

bv? = �bv?

2

⇣v

�2th,↵@tv

2th,↵ + J

�1r⇠

⇣bv|| � b

x

⌘v

�1th,↵@⇠v

2th,↵

⌘�

bv||bv?vth,↵r


Energy conservation in inertial terms is also critical for long-term accuracy

x2 4 6 8

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t = 1e-08t = 0.99858t = 2.9986t = 38.0856

t5 10 15 20 25 30 35

ener

gy c

ons

10-4

10-3

10-2

10-1

100

101 x2 4 6 8

T

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t = 1e-08t = 0.99143t = 3.005t = 38.0656

t5 10 15 20 25 30 35

ener

gy c

ons

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Correct equilibrium achieved

Analytical equilibrium

Wrong equilibrium due to numerical heating


vth adaptivity enables realistic simulations of multispecies plasmas

•  D-e-α, 3 species thermalization problem

•  Resolution with static grid:

•  Resolution with adaptivity and asymptotics:

•  Mesh savings of

Nv ⇠ 2

✓vth,e,1vth,D,0

◆2

= 140000⇥ 70000

Nv = 128⇥ 64

⇠ 106


Summary

•  Developed a fully implicit, optimal Fokker-Planck solver

•  Features phase-space adaptivity, optimal implicit time-stepping, and exact conservation properties –  MG preconditioned nonlinear solver –  Exact transformation of FP equation (no remapping) –  Careful conservative discretization (next talk)

•  As a result, we save many orders of magnitude in total simulation

time


Current and Future Focus

•  Both Lagrangian mesh and spherical geometry are implemented and being characterized

•  Performing extensive verification campaign with shock propagation in planar and spherical geometry (next talk)

•  Carry out planar geometry simulation campaign (this FY)

•  Prepare for spherical ICF simulations (next FY)


BACKUP SLIDES


Questions?


How does ICF work?

Laser Irradiation

fuel

capsule

compression to high T, ρ

Surfaceblowoff Rocket force


Rosenbluth-Fokker-Planck collision operator: simultaneous conservation of mass, momentum,

and energy


Rosenbluth-FP collision operator: conservation properties results from symmetries

m↵

�⌦v2, C↵�

↵~v

= �m�

�⌦v2, C�↵

↵~v

D~v, ~J�↵,G � ~J↵�,H

E

~v= 0)

Energy

h1, C↵�i~v = 0 )Mass

~J↵�,G � ~J↵�,H��~@v

= 0

and

br2

v,↵

"

bG��

✓

vth,�vth,↵

◆

4

#

| {z }

bG↵�

= bH↵� ,

which gives:

bG↵� = bG��

✓

vth,�vth,↵

◆

4

. (2.5.8)

Introducing these terms in the collision operator, and noting that the normalized collision operator is:

bC↵� = v3th,↵C↵� , (2.5.9)

we find:

bC↵� = v3th,↵�↵�

v3th,↵

1

vth,↵brv ·

"

1

v2th,↵brvbrv

vth,�

✓

vth,↵vth,�

◆

4

bG↵�

!

· 1

vth,↵brvbf↵ � m↵

m�

bf↵brv

vth,↵

bH↵�

vth,�

✓

vth,↵vth,�

◆

2

!#

,

which gives,

bC↵� =�↵�

v3th,�brv ·

brvbrvbG↵� · brv

bf↵ � m↵

m�

bf↵ brvbH↵�

�

. (2.5.10)

The above expression is obtained with: br2

vbH↵� = �8⇡ bf�

⇣

bv� = bv↵vth,↵

vth,�

⌘

or bH↵� = bH��

⇣

vth,�

vth,↵

⌘

2

, we obtain

br2

vbG↵� = bH↵� or bG↵� = bG��

⇣

vth,�

vth,↵

⌘

4

.

2.6 Treatment of Rosenbluth Potentials between Cross SpeciesVelocity Space

2.7 Conservative Discretization

We develop a mass, momentum, and energy conserving discretization for the Fokker-Planck operator.

2.7.1 Discrete Mass Conservation Scheme

First, recall the continuum mass conservation statement,

h1, C↵�iv =D

1,rv ·h

~JG,↵� + ~JH,↵�

iE

v=

h

1, ~JG,↵� + ~JH,↵�

i

@v�⇠⇠⇠⇠⇠⇠⇠⇠⇠⇠D

0, ~JG,↵� + ~JH,↵�

E

v. (2.7.1)

Here, the surface integral requires the evaluation of the collisional fluxes at the velocity boundaries to bezero. In the discrete, we do the exactly identical treatment by numerically setting the fluxes to zero at theboundary.

2.7.2 Discrete Momentum Conservation Scheme

For momentum conservation, the following relation must hold:

m↵ h~v, C↵�i~v = �m� h~v, C�↵i~v . (2.7.2)

8

)Momentum

D1, Jk

↵�,G � Jk�↵,H

E

~v= 0

C↵� = �↵�rv ·h~J↵�,G � m↵

m�

~J↵�,Hi


2V Rosenbluth-FP collision operator: numerical conservation of energy •  The symmetry to enforce is:

•  Due to discretization error:

•  Introduce a constraint coefficient:

D~v, ~J�↵,G � ~J↵�,H

E

~v= 0

D~v, ~J�↵,G � ~J↵�,H

E

~v= O (�v)

D~v, ��↵ ~J�↵,G � ~J↵�,H

E

~v= 0 ��↵ =

D~v, ~J↵�,H

E

~vD~v, ~J�↵,G

E

~v

= 1 +O (�v)

C↵� = �↵�rv ·�↵� ~J↵�,G � m↵

m�

~J↵�,H

�


•  Simultaneous conservation of momentum and energy: with:

2V Rosenbluth-FP collision operator: numerical conservation of momentum+energy

Energy

Momentum ⌘↵�

=

�↵� + ✏||,↵� 0

0 �↵�

�

�↵� =

D~v, ~JH,�↵

E

~v� ✏+↵�,||

D~v, ~JG,↵�

E+1

~v�~uD~v, ~JG,↵�

E

~v

✏↵� =

8<

:

✏�||,↵� = 0 if v|| � uavg,||,↵� 0

✏+||,↵� =h1,JH,�↵,||i~v��↵�h1,JG,↵�,||i~v

h1,JG,↵�,||i+1~v�~uavg,↵�

else

9=

;

C↵� = �↵�rv ·⌘↵�

· ~J↵�,G � m↵

m�

~J↵�,H

�


Numerical Results: Single-species initial random distribution thermalizes to a Maxwellian


Numerical Results: Conservation properties enforced down to nonlinear convergence tolerance

time0 0.01 0.02 0.03 0.04 0.05

∆M

10-16

10-14

10-12

10-10

10-8

10-6

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆I

10-12

10-10

10-8

10-6

10-4

10-2

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆U

10-15

10-10

10-5

100

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆S

10-2

10-1

100

101

ϵr=10-2

ϵr=10-4

ϵr=10-6


Vlasov equation: Inertial term simultaneous conservation of mass, momentum, and energy


•  Focus on temporal inertial terms due to normalization wrt vth(r,t) (OD):

•  Mass conservation can be trivially shown by 0th velocity space moment:

FP equation with adaptivity in velocity space: Temporal inertial terms

v2th@n↵

@t= 0

and

⌘↵�,? =

⇣

vth,�

vth,↵

⌘

5

⌧

~bv,~bJ�↵,G

�

~bv�

D

bv||, ⌘↵�,|| bJ↵�,H,||

E

~bvD

bv?, bJ↵�,H,?

E

~bv

. (2.7.30)

2.8 Conservative Discretization with Velocity Space Adaptivity

We discuss the development of a conservative discretization for the 0D conservation equation with velocityspace adaptivity. We begin by developing an independent mass, momentum, and energy conserving scheme,then develop a mass and momentum scheme, and finally develop a simultaneous mass, momentum, andenergy conserving discretization scheme.

2.8.1 Velocity Space Adaptivity: Exact Mass Conservation

We show that mass conservation is trivially enforced for the new conservation equation shown in equation(2.4.1) for a 0D case. Ignoring the collision operator (as can be shown to be play no role in the proof of

conservation properties with mesh adaptivity), and expandingpgv, v||, v?, and using v = vth,↵bv; f↵ = bf/v3th,↵,

we obtain:

@ bf↵@t

� 1

2

@lnT↵

@t

✓

@

@bv||bv|| +

@

@bv?bv?

◆

bf↵

�

= 0, (2.8.1)

which can be simplified as:

@ bf↵@t

� 1

2

@lnT↵

@tbrv ·

h

~bv bf↵

i

= 0. (2.8.2)

Since, @tlnT↵ = T�1

↵ @tT↵, and T↵ ⌘ v2th,↵, by multiplying the entire equation by v2th,↵ (the motive will beclear in a while) we obtain:

v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0. (2.8.3)

Mass conservation can be shown by taking the 0th moment of the above equation:

v2th,↵

Z

dbv

(

@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

)

= v2th,↵

⇢

@n↵

@t �⇢

��h

1,~bv bf↵i

@v��D

0,~bv bf↵E

v

��

= v2th,↵@n↵

@t = 0. (2.8.4)

Mass conservation is enforced as long as the divergence operator in velocity space is discretized such that thevelocity space flux at the boundary is set to zero.

2.8.2 Velocity Space Adaptivity: Exact Momentum Conservation

We derive the exact momentum conservation scheme of the new conservation equation shown in equation(2.4.1) for a 0D case. We begin with equation (2.8.3),

v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0.

Using chain rules,

v2th,↵@tbf↵ = vth,↵

h

@t

⇣

vth,↵ bf↵

⌘

� bf↵@tvth,↵

i

,

12


⌧bv2, bf↵ +

1

2brv ·

⇣~bv bf↵

⌘�

~v

= 0

Find symmetry in continuum and enforce via using discrete nonlinear constraints (similar to collisions)

and

⌘↵�,? =

⇣

vth,�

vth,↵

⌘

5

⌧

~bv,~bJ�↵,G

�

~bv�

D

bv||, ⌘↵�,|| bJ↵�,H,||

E

~bvD

bv?, bJ↵�,H,?

E

~bv

. (2.7.30)






we obtain:

@ bf↵@t

� 1

2

@lnT↵

@t

✓

@

@bv||bv|| +

@

@bv?bv?

◆

bf↵

�

= 0, (2.8.1)


@ bf↵@t

� 1

2

@lnT↵

@tbrv ·

h

~bv bf↵

i

= 0. (2.8.2)



v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0. (2.8.3)


v2th,↵

Z

dbv

(

@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

)

= v2th,↵

⇢

@n↵

@t �⇢

��h

1,~bv bf↵i

@v��D

0,~bv bf↵E

v

��

= v2th,↵@n↵

@t = 0. (2.8.4)




v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0.

Using chain rules,


h

@t

⇣

vth,↵ bf↵

⌘

� bf↵@tvth,↵

i

,

12

@U↵

@t= 0

and

@tv2

th,↵ = 2vth,↵@tvth,↵,

we obtain,

vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

~bv bf↵

⌘io

= 0. (2.8.5)

Momentum conservation can be shown by taking the first velocity moment of the above equation:

vth,↵

Z

dbv~bvn

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

~bv bf↵

⌘io

=

vth,↵

⇢

@t (n~u↵)� @tvth,↵

n~bu↵ �⇢

��h

~bv,~bv bf↵

i

@v�

D

1,~bv bf↵E

v

��

= @t (n~u↵) = 0. (2.8.6)

Here, the first velocity space moment of bf↵ and brv ·⇣

~bv bf↵

⌘

must cancel out. However, in the discrete, due to

discretization error, this relation is not satisfied (in general). In order to enforce this property in the discrete,we introduce ⌥

t,↵into the conservation equation:

vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

= 0. (2.8.7)

Here, ⌥t,↵

is defined as:

⌥t,↵

=

⌥||,↵ 00 ⌥?,↵

�

=

2

6

4

� hv||, bf↵ivhv||,brv·(~bv bf↵)i

v

0

0 � hv?, bf↵iv

hv?,brv·(~bv bf↵)iv

3

7

5

. (2.8.8)

The ⌥t,↵

factor enforces a component wise momentum conservation, however, ⌥?,↵ = 1 due to cylindrical

symmetry (i.e. perpendicular component of momentum is automatically conserved due to symmetry).

2.8.3 Velocity Space Adaptivity: Exact Energy Conservation


v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0.

Using the chain rule,

v2th,↵@tbf↵ = @t

⇣

v2th,↵bf↵

⌘

+ bf↵@tv2

th,↵, (2.8.9)

we obtain:

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

"

bf↵ +brv

2·⇣

~bv bf↵

⌘

#

= 0. (2.8.10)

Energy conservation can be shown by taking the trace of the 2nd velocity moment of the above equation:

Z

dbvbv2

2

(

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

"

bf↵ +brv

2·⇣

~bv bf↵

⌘

#)

=

@t

⇣

v2th,↵ bU↵

⌘

� @tv2

th,↵

"

bU↵ +⇠⇠⇠⇠⇠⇠⇠⇠⇠⇠"

bv2

2,brv

2·⇣

~bv bf↵

⌘

#

@v

�*

~bv,

~bv

2bf↵

+

v

#

=

@t

⇣

v2th,↵bU↵

⌘

� @tv2

th,↵

h

��bU↵ ��bU↵

i

= 0. (2.8.11)

13

The important property one must enforce is that the 2nd moment of bf↵ andbrv

2

·⇣

~bv bf↵

⌘

must cancel each

other out. In the discrete, this is generally not possible. In order to enforce exact energy conservation, wemodify equation (2.8.10) as:

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

h

bf↵ +�t,↵2

brv ·⇣

~bv bf↵

⌘i

= 0. (2.8.12)

Here, �t,↵ is the constraint coe�cient that enforces exact cancellation between the 2nd moments of bf↵ and�t,↵

2

brv ·⇣

~bv bf↵

⌘

and is defined as:

�t,↵ = �

D

bv2

2

, bf↵

E

bvD

bv2

2

, 1

2

brv ·⇣

~bv bf↵

⌘E

bv

. (2.8.13)

2.8.4 Velocity Space Adaptivity: Exact Mass and Momentum Conservation

Up to this point, we have discussed the development of a separate mass, momentum, and energy conservingscheme for a velocity space adaptive mesh scheme. We now extend the discretization scheme to a simultaneousmass and moment conserving scheme. In the development of the exact momentum conserving scheme, we

recognized that in the discrete, the 1st velocity moments of bf↵ and brv ·⇣

~bv bf↵

⌘

may not cancel exactly, hence

introducing the ⌥t,↵

factor in brv ·⇣

~bv bf↵

⌘

. In a similar spirit if one takes the 0th velocity moment of equation

(2.8.7), one obtains:

vth,↵ [@t (vth,↵n↵)� @tvth,↵n↵] = 0. (2.8.14)

In the continuum, we can perform the following manipulation,

@t (vth,↵n↵) = vth,↵@tn↵ + @tvth,↵n↵. (2.8.15)

Substituting this expression into equation (2.8.14), we obtain:

v2th,↵@tn↵ = 0, (2.8.16)

which is precisely a statement of mass conservation. However, in order for this expression to take place, werequire the chain rule shown in equation (2.8.15) to hold. In the discrete, the chain rule will not be satisfiedexactly. In order to enforce the chain rule exactly, we modify equation (2.8.7) as:

vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+ ⌘t,↵ = 0. (2.8.17)

Here, ⌘t,↵ is a discrete consistency source which is defined as:

⌘t,↵ (v) =n

v2th,↵@tbf↵ � @tv

2

th,↵brv ·

⇣

~bv bf↵

⌘o

�vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

~bv bf↵

⌘io

, (2.8.18)

which acts to enforce the chain rule shown in equation (2.8.15) exactly in the discrete. Note that unlike the⌥

t,↵factor which is a global (moment) quantity, ⌘t,↵ is a local term in phase-space (a function of v). With

⌘t,↵ defined as a source in the conservation equation, this does not rigorously enforce momentum conservationin the discrete. In order to enforce momentum conservation, we redefine ⌥

t,↵such as to absorb the new

truncation error arising from introducing ⌘t,↵:

⌥t,↵

=

⌥||,↵ 00 ⌥?,↵

�

=

2

6

4

�vth,↵@tvth,↵hv||, bf↵iv+hbv||,⌘t,↵iv

vth,↵@tvth,↵hv||,brv·(~bv bf↵)iv

0

0�vth,↵@tvth,↵hv?, bf↵i

v+hbv?,⌘t,↵iv

vth,↵@tvth,↵hv?,brv·(~bv bf↵)iv

3

7

5

. (2.8.19)

14

Dbv2, bf↵ +

�t,↵2

brv ·⇣~bv bf↵

⌘E

~v= 0

Rewrite as:

Energy conservation shown from 2nd velocity moment:

This property relies on:

This property must be enforced numerically:


We prove that equation (2.8.17) will enforce exact mass and momentum conservation in the discrete.Taking the 1st velocity moment of equation (2.8.17):

vth,↵@t

⇣

vth,↵n~bu↵

⌘

� vth,↵@tvth,↵

Z

dbv~bvh

bf↵ + brv ·⇣

~bv bf↵

⌘i

+

Z

dbv~bv⌘t,↵| {z }

= 0 from ⌥

t,↵

= 0, (2.8.20)

momentum conservation is shown. Now, taking the 0th moment, we obtain:

((((((((vth,↵@t (vth,↵n↵)�(((((((

vth,↵@tvth,↵n↵ +n

v2th,↵@tn↵

o

�⇠⇠⇠vth,↵�

⇠⇠⇠⇠⇠⇠@t (vth,↵n↵)�⇠⇠⇠⇠⇠@tvth,↵n↵

(2.8.21)

= v2th,↵@tn↵ = 0, (2.8.22)

mass conservation is shown. We also show that the addition of ⌘t,↵ does not break local mass conservationproperty. By substituting equation (2.8.18) into equation (2.8.17), we obtain:

vth,↵

n

⇠⇠⇠⇠⇠⇠@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

◆◆bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+n


2

th,↵brv ·

⇣

~bv bf↵

⌘o

�

vth,↵

n

⇠⇠⇠⇠⇠⇠@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

◆◆bf↵ + brv ·⇣

~bv bf↵

⌘io

=


2

th,↵brv ·

⇣

~bv bf↵

⌘

� vth,↵@tvth,↵ brv ·h⇣

⌥t,↵

� 1⌘

~bv bf↵

i

, (2.8.23)

which is in a purely conservative form.

2.8.5 Velocity Space Adaptivity: Exact Mass, Momentum, and Energy Conser-vation

Finally, we develop a simultaneous mass, momentum, and energy conserving scheme with velocity spaceadaptivity. We perform a recursive enslavement of the mass and momentum conserving equation (equation(2.8.17)) into the energy conserving equation (equation (2.8.12)), by introducing a discrete consistency source,⇠t,↵:

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

h

bf↵ + �t,↵ brv ·⇣

~bv bf↵

⌘i

+ ⇠t,↵ = 0. (2.8.24)

with,

⇠t,↵ = vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+ ⌘t,↵

�(

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

"

bf↵ +brv

2·⇣

~bv bf↵

⌘

#)

, (2.8.25)

and �t,↵ redefined to take into account of ⇠t,↵ as:

�t,↵ =�@tv

2

th,↵

D

bv2

2

, bf↵

E

v+D

bv2

2

, ⇠t,↵

E

v

@tv2th,↵

D

bv2

2

, brv ·⇣

~bv bf↵

⌘E

v

(2.8.26)

Here, ⌘t,↵ is defined in equation (2.8.18), and ⌥t,↵

is defined as:

⌥t,↵

=

⌥||,↵ 00 ⌥?,↵

�

, (2.8.27)

where,

⌥t,↵,|| =�vth,↵@tvth,↵

D

v||, bf↵

E

v� (�t,↵ � 1) @tv2th,↵

D

bv||,brv

2

·⇣

~bv bf↵

⌘E

v+⌦

bv||, ⌘t,↵↵

v

vth,↵@tvth,↵

D

v||, brv ·⇣

~bv bf↵

⌘E

v

, (2.8.28)

15

The important property one must enforce is that the 2nd moment of bf↵ andbrv

2

·⇣

~bv bf↵

⌘

must cancel each

other out. In the discrete, this is generally not possible. In order to enforce exact energy conservation, wemodify equation (2.8.10) as:

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

h

bf↵ +�t,↵2

brv ·⇣

~bv bf↵

⌘i

= 0. (2.8.12)

Here, �t,↵ is the constraint coe�cient that enforces exact cancellation between the 2nd moments of bf↵ and�t,↵

2

brv ·⇣

~bv bf↵

⌘

and is defined as:

�t,↵ = �

D

bv2

2

, bf↵

E

bvD

bv2

2

, 1

2

brv ·⇣

~bv bf↵

⌘E

bv

. (2.8.13)

2.8.4 Velocity Space Adaptivity: Exact Mass and Momentum Conservation

Up to this point, we have discussed the development of a separate mass, momentum, and energy conservingscheme for a velocity space adaptive mesh scheme. We now extend the discretization scheme to a simultaneousmass and moment conserving scheme. In the development of the exact momentum conserving scheme, we

recognized that in the discrete, the 1st velocity moments of bf↵ and brv ·⇣

~bv bf↵

⌘

may not cancel exactly, hence

introducing the ⌥t,↵

factor in brv ·⇣

~bv bf↵

⌘

. In a similar spirit if one takes the 0th velocity moment of equation

(2.8.7), one obtains:

vth,↵ [@t (vth,↵n↵)� @tvth,↵n↵] = 0. (2.8.14)

In the continuum, we can perform the following manipulation,

@t (vth,↵n↵) = vth,↵@tn↵ + @tvth,↵n↵. (2.8.15)

Substituting this expression into equation (2.8.14), we obtain:

v2th,↵@tn↵ = 0, (2.8.16)

which is precisely a statement of mass conservation. However, in order for this expression to take place, werequire the chain rule shown in equation (2.8.15) to hold. In the discrete, the chain rule will not be satisfiedexactly. In order to enforce the chain rule exactly, we modify equation (2.8.7) as:

vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+ ⌘t,↵ = 0. (2.8.17)

Here, ⌘t,↵ is a discrete consistency source which is defined as:

⌘t,↵ (v) =n


2

th,↵brv ·

⇣

~bv bf↵

⌘o

�vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

~bv bf↵

⌘io

, (2.8.18)

which acts to enforce the chain rule shown in equation (2.8.15) exactly in the discrete. Note that unlike the⌥

t,↵factor which is a global (moment) quantity, ⌘t,↵ is a local term in phase-space (a function of v). With

⌘t,↵ defined as a source in the conservation equation, this does not rigorously enforce momentum conservationin the discrete. In order to enforce momentum conservation, we redefine ⌥

t,↵such as to absorb the new

truncation error arising from introducing ⌘t,↵:

⌥t,↵

=

⌥||,↵ 00 ⌥?,↵

�

=

2

6

4

�vth,↵@tvth,↵hv||, bf↵iv+hbv||,⌘t,↵iv

vth,↵@tvth,↵hv||,brv·(~bv bf↵)iv

0

0�vth,↵@tvth,↵hv?, bf↵i

v+hbv?,⌘t,↵iv

vth,↵@tvth,↵hv?,brv·(~bv bf↵)iv

3

7

5

. (2.8.19)

14

•  Rewrite as:

All conservation law can be enforced via recursive application of chain rule

and

⌘↵�,? =

⇣

vth,�

vth,↵

⌘

5

⌧

~bv,~bJ�↵,G

�

~bv�

D

bv||, ⌘↵�,|| bJ↵�,H,||

E

~bvD

bv?, bJ↵�,H,?

E

~bv

. (2.7.30)






we obtain:

@ bf↵@t

� 1

2

@lnT↵

@t

✓

@

@bv||bv|| +

@

@bv?bv?

◆

bf↵

�

= 0, (2.8.1)


@ bf↵@t

� 1

2

@lnT↵

@tbrv ·

h

~bv bf↵

i

= 0. (2.8.2)



v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0. (2.8.3)


v2th,↵

Z

dbv

(

@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

)

= v2th,↵

⇢

@n↵

@t �⇢

��h

1,~bv bf↵i

@v��D

0,~bv bf↵E

v

��

= v2th,↵@n↵

@t = 0. (2.8.4)




v2th,↵@ bf↵@t

�@tv

2

th,↵

2brv ·

⇣

~bv bf↵

⌘

= 0.

Using chain rules,


h

@t

⇣

vth,↵ bf↵

⌘

� bf↵@tvth,↵

i

,

12

Truncation error

We prove that equation (2.8.17) will enforce exact mass and momentum conservation in the discrete.Taking the 1st velocity moment of equation (2.8.17):

vth,↵@t

⇣

vth,↵n~bu↵

⌘

� vth,↵@tvth,↵

Z

dbv~bvh

bf↵ + brv ·⇣

~bv bf↵

⌘i

+

Z

dbv~bv⌘t,↵| {z }

= 0 from ⌥

t,↵

= 0, (2.8.20)

momentum conservation is shown. Now, taking the 0th moment, we obtain:

((((((((vth,↵@t (vth,↵n↵)�(((((((

vth,↵@tvth,↵n↵ +n

v2th,↵@tn↵

o

�⇠⇠⇠vth,↵�

⇠⇠⇠⇠⇠⇠@t (vth,↵n↵)�⇠⇠⇠⇠⇠@tvth,↵n↵

(2.8.21)

= v2th,↵@tn↵ = 0, (2.8.22)

mass conservation is shown. We also show that the addition of ⌘t,↵ does not break local mass conservationproperty. By substituting equation (2.8.18) into equation (2.8.17), we obtain:

vth,↵

n

⇠⇠⇠⇠⇠⇠@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

◆◆bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+n


2

th,↵brv ·

⇣

~bv bf↵

⌘o

�

vth,↵

n

⇠⇠⇠⇠⇠⇠@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

◆◆bf↵ + brv ·⇣

~bv bf↵

⌘io

=


2

th,↵brv ·

⇣

~bv bf↵

⌘

� vth,↵@tvth,↵ brv ·h⇣

⌥t,↵

� 1⌘

~bv bf↵

i

, (2.8.23)

which is in a purely conservative form.

2.8.5 Velocity Space Adaptivity: Exact Mass, Momentum, and Energy Conser-vation

Finally, we develop a simultaneous mass, momentum, and energy conserving scheme with velocity spaceadaptivity. We perform a recursive enslavement of the mass and momentum conserving equation (equation(2.8.17)) into the energy conserving equation (equation (2.8.12)), by introducing a discrete consistency source,⇠t,↵:

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

h

bf↵ + �t,↵ brv ·⇣

~bv bf↵

⌘i

+ ⇠t,↵ = 0. (2.8.24)

with,

⇠t,↵ = vth,↵

n

@t

⇣

vth,↵ bf↵

⌘

� @tvth,↵

h

bf↵ + brv ·⇣

⌥t,↵

~bv bf↵

⌘io

+ ⌘t,↵

�(

@t

⇣

v2th,↵bf↵

⌘

� @tv2

th,↵

"

bf↵ +brv

2·⇣

~bv bf↵

⌘

#)

, (2.8.25)

and �t,↵ redefined to take into account of ⇠t,↵ as:

�t,↵ =�@tv

2

th,↵

D

bv2

2

, bf↵

E

v+D

bv2

2

, ⇠t,↵

E

v

@tv2th,↵

D

bv2

2

, brv ·⇣

~bv bf↵

⌘E

v

(2.8.26)

Here, ⌘t,↵ is defined in equation (2.8.18), and ⌥t,↵

is defined as:

⌥t,↵

=

⌥||,↵ 00 ⌥?,↵

�

, (2.8.27)

where,

⌥t,↵,|| =�vth,↵@tvth,↵

D

v||, bf↵

E

v� (�t,↵ � 1) @tv2th,↵

D

bv||,brv

2

·⇣

~bv bf↵

⌘E

v+⌦

bv||, ⌘t,↵↵

v

vth,↵@tvth,↵

D

v||, brv ·⇣

~bv bf↵

⌘E

v

, (2.8.28)

15


Cold speciesessentially a deltafunction on hotspecies' mesh

Hot

Cold Cold

Hot

Hot species' meshis too coarse forinterpolation oncold species' mesh

Asymptotic Formulation of Interspecies Collisions for vth,f >> vth,s

Argument of exp: vf/vth,s >> 1 Argument of exp: vs/vth,f << 1

[5] W.T. Taitano, JCP accepted (2016)


1D2V Mach 5 shock agrees well with known reference solution

x0 20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

nuTiTe

M=5 planar steady state shock solution (SSS) comparison between iFP (solid line) and reference solution from [10] (open circles). [10] F. Vidal et al., PoF B, 3182 (1993)


Lagrangian mesh tracks shock

x0 500 1000

T

0

0.2

0.4

0.6

0.8

1

1.2 t = 100.132

LagrangianShifted SSS

x0 500 1000

T

0

0.2

0.4

0.6

0.8

1

1.2 t = 199.818

LagrangianShifted SSS

Test of Lagrangian mesh capability for a M=5 shock in lab frame. A good agreement in solutions between Lagrangian mesh and SSS is achieved.


Spherical geometry is being tested

Test of spherical shock convergence. Shock reflection is observed.

radius0 100 200 300 400 500 600

n

0

1

2

3

4

5

6

7

8t = 0.47933t = 3.358t = 24.1781t = 104.178

radius0 100 200 300 400 500 600

u

-2.5

-2

-1.5

-1

-0.5

0

0.5t = 0.47933t = 3.358t = 24.1781t = 104.178

radius0 100 200 300 400 500 600

T

0

0.5

1

1.5

2

2.5

3

3.5 i t = 0.47933e t = 0.47933i t = 3.358e t = 3.358i t = 24.1781e t = 24.1781i t = 104.178e t = 104.178

Incoming shock

Reflected shock


More moments for Lagrangian mesh


Stably and accurately integrate with Δt >> τcol

Test of implicit solver with D-Al interface problem with Δt = 4x104 τcol.

ξ-3 -2 -1 0 1 2 3

ρ/ρ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 DiFP t = 6.0493e-09[ps]AliFP t = 6.0493e-09[ps]DiFP t = 19.828[ps]AliFP t = 19.828[ps]DiFP t = 24.72[ps]AliFP t = 24.72[ps]DMolvig PRLAlMolvig PRL


Velocity space adaption does not help for interspecies collisions

•  For electron-ion collisions, vth,e/vth,i >> 1

•  Similarly, for α-ion collisions, vth,α/vth,I >> 1

•  Very stringent mesh resolution requirements if determining

potentials via:

•  Mesh requirement grows as: •  Velocity space adaption helps ONLY for self-species, but not for

interspecies! We need asymptotics

v||

v┴ Fast Grid

Slow Grid

Region of rapidlyvarying slowpotential structure.

r2vHj (~v) = �8⇡fj (~v) r2

vGj (~v) = Hj (~v)

Ndv /

✓vth,fvth,s

◆d


Asymptotics: Fast on slow

•  Asymptotic potentials diverge as v à 0 •  Solution: Near v = 0, we coarse-grain slow potentials to

modify singularity

hHssi⌦TCR=

R⌦TCR

Hssd3v

R⌦TCR

d3v

r2vGfs = Hfs

Hfs =

⇢Hfs,asy if ~vf 62 ⌦TCR

hHssi⌦TCRelse

�


Numerical Results: Two-species thermal and momentum equilibration

time0 10 20 30

T

0

5

10

15

20

25

30T1T2

time0 10 20 30

u-4

-2

0

2

4 u1u2


Numerical Results: Conservation is enforced down to nonlinear tolerance

time0 10 20 30

∆M

10-14

10-12

10-10

10-8

10-6

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 10 20 30

∆I

10-15

10-10

10-5

100

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 10 20 30

∆U

10-15

10-10

10-5

100

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 10 20 30

∆S

10-5

10-4

10-3

10-2

10-1

ϵr=10-2

ϵr=10-4

ϵr=10-6


Non-Maxwellian effects for hot-cold proton thermalization

Initial Condition With positivity Without positivity


Our approach: nonlinearly implicit, adaptive, conserving Rosenbluth-VFP (cont.) •  Why Rosenbluth-VFP? •  Linearization and/or the use of asymptotic

approximations of collision operators are not warranted in these systems: –  Linearization is not allowed because deviations from

Maxwellian can be significant (Kn ~ 0.1 in fuel, boundary effects at material boundaries)

–  Ion thermal velocities are comparable to Gamow peak at bang time ( )

–  Multiple ion species with similar mass and temperatures

•  Nonlinear Rosenbluth-VFP allows for first-principles simulations

vth/vG ⇠ 1/p3


Static uniform mesh is not practical

•  Temperature disparities of 104 (for a single species)

•  10 points per vth

•  For multiple species (e.g. alpha-D/T)

•  Static spatial resolution:

) �v ⇠ vminth /10

vmax

th

/vmin

th

⇠ 100

Nv

⇠ (vmax

th

/�v)2 ⇠ 106

N⇤v ⇠ Nv30

2 ⇠ 109

N ⇠ N⇤v ⇥Nr ⇠ 1012 � 1013

Nr ⇠ 103 � 104


Challenges of Rosenbluth-VFP: temporal stiffness. Explicit method are impractical

•  Collision frequency dictates explicit time step:

•  Typical conditions at end of compression phase:



Nr

⇠ 103.


i


⌧i

⇡ R

vi

⇠ 5 ns.


�tcoll

exp

⇠ �v2

D⇠ �v2

v2

th

⌫coll

.


ni

⇡ 200 g/cm3 ⇡ 5⇥ 1025 cm�3 , Ti

⇡ 2.5 keV.


⇡ 10vmin

th


⌫coll

= 4.8⇥ 10�8

✓

mp

mi

◆

1/2 ni

(cc�1) ln⇤T

i

(eV )3/2

s�1 ⇡ 105 ns�1,


�tcoll

exp

⇠ 110

✓

�v

vmin

th

◆

2

⌫�1

coll

⇠ 10�9 ns,


i


i



max

⇠ 10vmax

th

:

�tstrexp

⇠ �r

vmax

⇠ R

Nr

10vmax

th

.

For Tmax


th


⇠ 103, wefind:

�tstrexp

⇠ 10�4 � 10�5 ns.




Nr

⇠ 103.


i


⌧i

⇡ R

vi

⇠ 5 ns.


�tcoll

exp

⇠ �v2

D⇠ �v2

v2

th

⌫coll

.


ni

⇡ 200 g/cm3 ⇡ 5⇥ 1025 cm�3 , Ti

⇡ 2.5 keV.


⇡ 10vmin

th


⌫coll

= 4.8⇥ 10�8

✓

mp

mi

◆

1/2 ni

(cc�1) ln⇤T

i

(eV )3/2

s�1 ⇡ 105 ns�1,


�tcoll

exp

⇠ 110

✓

�v

vmin

th

◆

2

⌫�1

coll

⇠ 10�9 ns,


i


i



max

⇠ 10vmax

th

:

�tstrexp

⇠ �r

vmax

⇠ R

Nr

10vmax

th

.

For Tmax


th


⇠ 103, wefind:

�tstrexp

⇠ 10�4 � 10�5 ns.


N�t ⇠ 1010


Our approach: nonlinearly implicit, adaptive, conserving Rosenbluth-VFP

•  Adaptive meshing to alleviate mesh requirements: –  Velocity space adaptivity via renormalization

(Nv~104-105)

–  Spatial adaptivity via Lagrangian mesh (Nr ~ 102)

•  Strict conservation properties (mass, momentum, energy).

v = v/vth(r, t)

N = Nv ⇥Nr ⇠ 106 � 107


Logo choices

OR

an optimal vlasov-fokker-planck solver for simulation of ... · outline • motivations ......

Documents