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圧縮性MHDに対する
ロバストな数値計算法の開発
第20回NEXT(数値トカマク)研究会
2015年1月13日(火) – 1月14日(水) 京都テルサ
三好 隆博
広島大学大学院理学研究科
Development of a robust scheme
for compressible MHD
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MHD code projects
For laboratory plasmas
Extended MHD model
Not designed for shock capturing
Project Developer MHD Scheme div B Grid
(US)
NIMROD Sovinec FEM(2D)+SP(1D) / implicit Diffusion Triangular
M3D-C1 Ferraro, Jardin C1 FEM / implicit Vec. pot. Triangular
(EU)
XTOR-2F Lütjens FD(1D)+SP(2D) / NK implicit ― Mag. Flux
(Japan)
MIPS Todo 4th FD / 4th RKG ― Cylindrical
MINOS Miura 8th Compact FD / 4th RKG ― Curvilinear
Any information or corrections are appreciated…
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Shocks in space plasmas
Coronal activities (Hinode)
Ubiquitous reconnection / jet (Hinode)
Magnetosphere (SCOPE)
Heliosphere (Artist’s image/NASA)
Jet from black hole (ATCA)
Petschek-type reconnection
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Motivation and objectives
In compressible MHD codes for laboratory plasmas,
time integration methods have been polished so as
to solve stiff problems.
But, those codes are not designed for shock
capturing that may be needed in the near future.
(e.g., HiFi code at PSI-Center)
Shocks and turbulence are universally observed in
space. Thus, the development of robust shock
capturing schemes has been highly progressed.
Current status and challenges of the shock
capturing scheme for MHD are presented with
emphasis on our results.
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Compressible MHD equations
Ideal MHD equations (Non-conservative form)
Various non-conservative forms can be
obtained using vector identities
equationinduction
equationadiabatic
motionofequation
equationcontinuity
:0,0
:0
:
:0
BBB
BJ
v
v
vvv
v
t
pp
t
pt
t
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Compressible MHD equations
Ideal MHD equations (Conservative form)
2
,22
1,0
0
0
0
0
222 Bpp
Bep
t
pet
e
pt
t
T
T
T
v
B
BBB
BB
BBI
onconservatiflux:
onconservatienergy:
onconservatimomentum:
onconservatimass:
vv
vv
vvv
v
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Shock capturing scheme
Non-conservative scheme
Based on non-conservative form
Converge to unphysical shock
Hou-LeFloch [1994]
Conservative scheme
Based on conservative form
Converge to physical shock
Lax-Wendroff [1960]
Harten [1980]
Difficult to preserve positivity
“Computational Tutorial: MHD”, Toth http://www.lorentzcenter.nl/lc/web/2011/441/presentations/Advanced_Toth.pdf
Conservative vs Non-conservative
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Shock capturing scheme
Non-conservative scheme
Finite difference method
Finite element method
Conservative scheme
Finite difference method
FD-WENO, Compact FD+LAD, etc.
Finite element method
RKDG, etc.
Finite volume method
MUSCL, FV-WENO, etc.
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Shock capturing scheme
1D finite volume method
Numerical flux
0
,,0 2/12/1
x
txuftxufu
dt
d
x
f
t
u iii
,,,,,, 2112/12/1 iiiiii uuuufftxuf
2/1ix 2/1ix
iu 1iu1iu 2iu2/1if2/1if
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Approximate Riemann solver
Approximate Riemann solver
Define piecewise constants
0dtdxdxdt
xtFU
FU
U
x
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Approximate Riemann solver
Approximate Riemann solver
Define piecewise constants
Solve local Riemann problems
0dtdxdxdt
xtFU
FU
U
x
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Riemann problem
Riemann problem = Shock tube problem
7-waves can be excited in 1D MHD system
(shock, expansion wave, compound wave)
LRtx UUUU ,;t
RU
FRFS /RDSRSS /CDRDFRFS / SRSS /
LU
x
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Approximate Riemann solver
Approximate Riemann solver
Define piecewise constants
Solve local Riemann problems
Average state variables
0dtdxdxdt
xtFU
FU
U
x
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Approximate Riemann solver
Approximate Riemann solver
Define piecewise constants
Solve local Riemann problems
Average state variables
Derive numerical fluxes from conservation laws
Depend on “quality” of approximate solutions!
0dtdxdxdt
xtFU
FU
U
x
0,; 2/12/112/12/1
n
ii
n
iii
x
x
n
i
n
ii txxdxt
xxi
i
FFUUUU
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Approximate Riemann solver
Standard approximate Riemann solver
Lax-Friedrichs scheme [Lax, 1950’s]
Godunov scheme [Godunov, 1959]
Rusanov scheme [Rusanov, 1961]
Roe scheme (HD) [Roe, 1981]
HLL scheme [Harten+, 1983]
Roe scheme (MHD) [Brio+, 1988]
HLLC scheme (HD) [Toro+, 1994; Batten+, 1997]
HLLC scheme (MHD) [Gurski, 2004; Li, 2005]
HLLD scheme (MHD) [Miyoshi+, 2005]
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HLL approximate Riemann solver
HLL Riemann solver [Harten+, 1983]
Conservation laws
2-waves approximation
CD/TD/RD cannot be resolved
LRS , :max./min. speeds
RULU
RFLF
RSLS
x
t
2/1i
U
00 * LRLLRRLR SSSSdtdx FFUUUFU
0,,min
0,,max
RRLLL
RRLLR
cucuS
cucuS
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HLL approximate Riemann solver
HLL Riemann solver [Harten+, 1983]
Conservation laws
2-waves approximation
CD/TD/RD cannot be resolved
LRS , :max./min. speeds
RULU
RFLF
RSLS
x
t
2/1i
U
00 *
FFUUFU RRRR SSdtdx
0,,min
0,,max
RRLLL
RRLLR
cucuS
cucuS
F
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HLLD approximate Riemann solver
HLLD Riemann solver [Miyoshi+, 2005]
Conservation laws
5-waves approximation
LRS , :fast magnetosonic wave
RULU
RFLF
RSLS
x
t
2/1i
TM pS ,
RU
RU
LU
LU
MS :entropy wave *
,LRS :Alfvén wave
*
RS*
LS MS
0,1
,
,,
1********
*
,
**
,
*
,
**
,
*
,,
*
,,
*
,,
LRLLRR
tS
tS
n
LRLRM
LRLRLRLRLRLRLRLRLRLR
SSdxtxt
S
SS
R
L
FFUUUFFUU
FFUUFFUU
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HLLD approximate Riemann solver
The HLLD Riemann solver
is constructed without eigenvectors
exactly resolves isolated CD/TD/RD/FS
preserves density and pressure positivities
High-efficiency! High-resolution! Robust!
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HLLD approximate Riemann solver
Established as a standard Riemann solver
Comparing numerical methods [Kritsuk+, 2011]
Athena (US), CANS+ (Japan), and many other
researches
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Challenges
Challenges to multi-D MHD scheme
Comparing numerical methods [Kritsuk+, 2011]
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Challenges to multi-D
Treatment of numerical magnetic monopole
Negative effect due to unphysical magnetic force
Need divergece-free/divergence-cleaning method!
(with correction) (without correction)
BBBBBBI 22B
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Challenges to multi-D
Treatment of numerical magnetic monopole
Can numerical simulations
preserve ∇・B = 0?
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Challenges to multi-D
Numerical shock instabilities
Odd-even decoupling
Carbuncle phenomena
(HLLD)
(HLLD-)
(Roe)
(HLLC-) (HLLC) (HLLD-) (HLLD)
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Challenges
Challenges to higher-order MHD scheme
Comparing numerical methods [Kritsuk+, 2011]
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Challenges to higher-order
Importance of higher-order methods
Error of nth-order method vs. Computational cost
0
x
ua
t
u
i
ii txuuN
L ,1
1
xtxu 2sin0,
se
ve
ral tim
es
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Challenges to higher-order
Godunov’s theorem
Any linear monotone scheme (non-oscillatory
scheme) can be at most first-order accurate
This statement suggests that higher-order non-
oscillatory scheme can be constructed as a
nonlinear scheme
TVD, MUSCL, PPM, WENO, etc.
Very-high-order WENO
(up to 17th-order)
[Gerolymos+, 2009]
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Challenges
Multi-dimensional higher-order divergence-free
scheme is one of the goals of shock capturing
scheme for MHD
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Summary
I have reported current status and challenges of
robust shock capturing schemes for MHD
The HLLD has been established as a standard
MHD solver in the field of astrophysics
Multi-D shock capturing scheme for MHD is one
of the challenges
Treatment of numerical magnetic monopole
Treatment of numerical shock instabilities
Higher-order shock capturing scheme for MHD is
one of the challenges
Study on shock capturing scheme for two-fluid /
extended MHD is now progressing…