Slide 1 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
Slide 2 U N C L A S S I F I E D
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
Slide 3 U N C L A S S I F I E D
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)
Slide 4 U N C L A S S I F I E D
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
Slide 5 U N C L A S S I F I E D
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)
Slide 6 U N C L A S S I F I E D
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
Slide 7 U N C L A S S I F I E D
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.
Slide 8 U N C L A S S I F I E D
• 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,↵
Slide 9 U N C L A S S I F I E D
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
Slide 10 U N C L A S S I F I E D
MG preconditioner keeps linear iterations bounded
Slide 11 U N C L A S S I F I E D
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
Slide 12 U N C L A S S I F I E D
Exact energy conservation in collision operator is critical for accuracy
• For demonstration, we consider electron-proton thermalization
• More details in Taitano’s talk
Slide 13 U N C L A S S I F I E D
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
Slide 14 U N C L A S S I F I E D
Adaptivity in physical space: Lagrangian mesh evolves with capsule
Slide 15 U N C L A S S I F I E D
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?
Slide 16 U N C L A S S I F I E D
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
Slide 17 U N C L A S S I F I E D
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
Slide 18 U N C L A S S I F I E D
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
Slide 19 U N C L A S S I F I E D
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
Slide 20 U N C L A S S I F I E D
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)
Slide 21 U N C L A S S I F I E D
BACKUP SLIDES
Slide 22 U N C L A S S I F I E D
Questions?
Slide 23 U N C L A S S I F I E D
How does ICF work?
Laser Irradiation
fuel
capsule
compression to high T, ρ
Surfaceblowoff Rocket force
Slide 24 U N C L A S S I F I E D
Rosenbluth-Fokker-Planck collision operator: simultaneous conservation of mass, momentum,
and energy
Slide 25 U N C L A S S I F I E D
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
Slide 26 U N C L A S S I F I E D
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
�
Slide 27 U N C L A S S I F I E D
• 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
�
Slide 28 U N C L A S S I F I E D
Numerical Results: Single-species initial random distribution thermalizes to a Maxwellian
Slide 29 U N C L A S S I F I E D
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
Slide 30 U N C L A S S I F I E D
Vlasov equation: Inertial term simultaneous conservation of mass, momentum, and energy
Slide 31 U N C L A S S I F I E D
• 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
Slide 32 U N C L A S S I F I E D
⌧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)
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
@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
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 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:
Slide 33 U N C L A S S I F I E D
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
v2th,↵@tbf↵ � @tv
2
th,↵brv ·
⇣
~bv bf↵
⌘o
�
vth,↵
n
⇠⇠⇠⇠⇠⇠@t
⇣
vth,↵ bf↵
⌘
� @tvth,↵
h
◆◆bf↵ + brv ·⇣
~bv bf↵
⌘io
=
v2th,↵@tbf↵ � @tv
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
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
• 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)
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
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
v2th,↵@tbf↵ � @tv
2
th,↵brv ·
⇣
~bv bf↵
⌘o
�
vth,↵
n
⇠⇠⇠⇠⇠⇠@t
⇣
vth,↵ bf↵
⌘
� @tvth,↵
h
◆◆bf↵ + brv ·⇣
~bv bf↵
⌘io
=
v2th,↵@tbf↵ � @tv
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
Slide 34 U N C L A S S I F I E D
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)
Slide 35 U N C L A S S I F I E D
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)
Slide 36 U N C L A S S I F I E D
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.
Slide 37 U N C L A S S I F I E D
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
Slide 38 U N C L A S S I F I E D
More moments for Lagrangian mesh
Slide 39 U N C L A S S I F I E D
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
Slide 40 U N C L A S S I F I E D
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
Slide 41 U N C L A S S I F I E 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
�
Slide 42 U N C L A S S I F I E D
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
Slide 43 U N C L A S S I F I E D
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
Slide 44 U N C L A S S I F I E D
Non-Maxwellian effects for hot-cold proton thermalization
Initial Condition With positivity Without positivity
Slide 45 U N C L A S S I F I E D
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
Slide 46 U N C L A S S I F I E D
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
Slide 47 U N C L A S S I F I E D
Challenges of Rosenbluth-VFP: temporal stiffness. Explicit method are impractical
• Collision frequency dictates explicit time step:
• Typical conditions at end of compression phase:
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.
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.
N�t ⇠ 1010
Slide 48 U N C L A S S I F I E D
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
Slide 49 U N C L A S S I F I E D
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