Structure-preserving geometric algorithms for plasma physics

Hong Qin1,2, Joshua W. Burby1, Philip J. Morrison3,

Yao Zhou1, and C. Leland Ellison1

1. Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543 2. School of Nuclear Science and Technology and Department of Modern Physics, University of

Science and Technology of China, Hefei, Anhui 230026, China 3. Physics Department, The University of Texas at Austin, Austin, TX 78712

The idea

Conventional simulation algorithms do not preserve the geometric structures of the physical systems, such as the local energy-momentum conservation laws and the symplectic structure.

Thus, numerical errors accumulate coherently with time and long-term simulation results are not reliable.

Structure-preserving geometric algorithms (will) solve this problem.

Symplectic and volume-preserving integrators for charged particles.

Variational & symplectic algorithms for the Vlasov-Maxwell system.

Structure-preserving geometric algorithm for MHD.

……

Geometric algorithms

built on the more fundamental field-theoretical formalism,

linked to the perfect form, i.e., the variational principle of physics.

The fact that the most elegant form of theory is also the most effective algorithm is philosophically satisfactory.

Theoretical models fundamentally are algorithms too. Good theoretical models should be built geometrically as well.

Vision: future numerical capabilities should be based on structure-preserving geometric algorithms

Geometric algorithm (e.g., symplectic) LaserNon-geometric algorithm (e.g., RK4) Flashlight

Geometric formalism and algorithms suitable for plasma physics

cannot be adapted from existing mathematical literature,

to be discovered by theoretical plasma physicists.

Fact: geometric algorithms are discovered by physicists.

Software facilities

Utilize structure-preserving geometric algorithms to maximize the benefit of next generation super-computing hardware.

Challenges and frontiers

Multiscale dynamics in a tokamak

Time-scale span: 1011

Direct numerical simulation impossible

83 10 Hz

43 10 Hz

3Device: 10 Hz

Heating: 108 – 102 Hz Transport: 104 – 10-3 Hz

Physics (as we know up to now) is harmonic oscillator

Algorithms

nth (4th) order Runge-Kutta methods

Long time non-conservation.

Errors add up coherently.

Carl Runge

(1856-1927) Martin Kutta (1867-1944)

( , )

1

1 1 2 3 4

4

12 2

6Error o

n n

n n

dxf t x

dtt t h

x x h k k k k

h

,,

,,

1

1 12 12 2

1 13 22 2

2 1 3

n n

n n

n n

n n

k f t x

k f t h x hk

k f t h x hk

k f t x hk

RK4 for pendulum: Errors add up coherently for 1 million time-steps

2

2 0 d x

xdt

Exact solutions

RK4

Can we do better than RK4? -- Symplectic Integrator

Feng (1983) Ruth (1983)

Conserves symplectic structure; Bound energy error globally

Application areas: Accelerator physics Planetary dynamics Nonlinear dynamics

1M time-step

Structure-preserving geometric algorithms for charged particles

Numerical result by RK4

RK4 for particles in tokamak – banana orbits go bananas

Exact banana orbit

4banana10 T

~ 6 burn bananaITER: 3 10T T

Variational symplectic integrator

1

0

212

tA Ldt

L u u B

A b X

[ ...,( ) ]discretize on

0 2 1t h h N h

1

0

1 1

1

1

N

d dk

d d k k k k

A A hL k k

L k k L u u

x x

0i d

( )1 1 0 1 2 3

1 1 0

d djk

d dk

L k k L k k jx

L k k L k ku

( )minimize w.r.t. k ku,x

discretized Euler-Lagrange Eq.

( ) ( )( )

1 1

1 1

,k k k k

k k

u u

u− −

+ +

, , → ,

x x

x

Variational symplectic integrator => perfect banana orbits

4banana10 TTransport reduction

by integration errors ~ 6

burn bananaITER: 3 10T T

Variational symplectic

RK4

[Qin08, Qin09, Li11, Squire12, Kraus13, Zhang14, Ellison15 ]

Why is Boris algorithm (1972) so good and what is better?

[Qin13, Webb14, He15, ZhangS15, Ellison15]

It is not symplectic, but volume-preserving (for non-relativistic particles)

[Relativistic volume-preserving algorithm, Zhang15]

Structure-preserving geometric algorithms for Vlasov-Maxwell systems

Vlasov-Maxwell system

Vlasov

Maxwell

( )

( )

( )

3

3

0

4 1

1

4

0

jj

j

j jj

j jj

ef t

t m c

e d p f tc c t

c t

e d pf t

v Bv E x p

x p

EB v x p

BE

E x p

B

Variational Symplectic:

[Squire12] Vlaov-Maxwell, geometric, DEC.

[Xiao13] Vlasov-Maxwell, smooth function.

[Evstatiev13] Vlasov-Poisson.

[Evstatiev14] Drift-kinetic.

Discrete symplectic structure:

[Qin15] Vlasov-Maxwell, canonical symplectic.

[Crouseilles15, Qin15b, He15b] Vlasov-Maxwell, splitting methods for the noncanonical Morrison-Marsden-Weinstein bracket.

[Burby15] Vlasov-Maxwell, Gyrokinetic (in progress).

[……]

Methods to geometrically discretize Vlasov-Maxwell system

Geometrically generalized Vlasov-Maxwell system --- A field theory

xI L

1( )

1( ) 4

2 xL x dA dA f

Valid for any .

Exact conservation propertie.

Allow physics models, approximations build into .

16d d d

fL pL

[Qin07 ]

Discretized Vlasov-Maxwell action

Action discretely gauge invariant ( invariant).A A df

Charge is conserved. Discrete Gauss’ law is exactly satisfied.

[Squire12] [Xiao13]

Theoretical formalism for Maxwell’s equations

4 1

1

4

0

jc c t

c t

EB

BE

EB *

0dF

d F J

1900 2000 1862

Which form is the best?

Vector calculus Scalar calculus Exterior calculus

The best form should be the most computable Discrete Exterior Calculus (DEC) Solver for Maxwell’s equations

Image from Desbrun06

0

*

dF

d F J

Discretizing Maxwell’s equations on discrete manifold guarantees exact conservation laws.

Discrete canonical symplectic structure for Vlasov-Maxwell field

{ , } { , }F G F G G F

F G f d d df f

xp x p xA Y A Y

2 2 21 1( , , ) ( ) ( )

2 2H f fd d d A Y p A x p Y A x

1 1

, ,N M

di Ji i i i J J J J

F G G F F G G FF G

X P X P A Y A Y

2 3

1 12 2

3

1 1

1( , , , ) ( ) ( ) ( )) ( ( ) d

2

1( ) ( ) ( ) ( ) d ,

2

M N

d i i J J i J J iJ i

M M

J J J JJ J

H t t W t

t W t W

X P A Y P A x x x X x

Y x x A x x x

[Qin15]

Fast local symplectic implicit algorithms are discovered

• No root researching or global matrix inversion are needed.

• Can carry out very large-scale plasma simulations with long-term precision and fidelity. For example: • Degrees of freedom=1012 and Number of time-steps=106 .

• Satisfy D2 local conservation laws in phase space that other

methods cannot satisfy.

Landau damping Phase mixing in velocity space

Structure-preserving geometric algorithms for MHD

Newcomb’s variational principle for MHD (1962)

3 30 0 0

0 0 0

0 0 0

0

0

d d / ,

/ / / ,

d d /

/ det

, ,

,i i i i i ij j

ij i j ij

x x J

p p p p J

B S B S B x B J

x x x J x

t t

x x v x

0 02 300 01

1( , , ) d

2 2( 1)ij ik j k

ij

x x B BpL x x

JJ

x xx

Difficulty: how to geometrically discretize in space?

Geometric discretization in space

[Zhou14]

3 0 3 2 3

3 3 22 0 2 2

2

, , | |( , , , ) , ,

8 1 2 | |[ ]p

L p v B

v B

DEC Eulerian labeling:

DEC Lagrangian labeling

3 0 3 2 30 0 0 0 0

3 3 22 2 20 0 0 0

0 01 2

, , | |( , ) ,

8 ( 1) 2 | |[ ]p

L x BJ

x x

0 00 0

( , ) vertex of t x

3 30 0

3 3 13 3 10 0 0

3 30 0

2 20 0

, , ,

, | |, | |,

, ,, , .

pp

B B

Variational integrators in time follow straightforwardly.

Variational methods for solving MHD

Symplectic and momentum-preserving.

Building in the advection equations without the error and dissipation.

Physically transparent. The moving mesh practically simulates the motion of fluid elements.

Application: spontaneous current sheet formation in ideal MHD.

Coalescence instability [Longcope93]

Unstable equilibrium Relaxed equilibrium with current sheet

Conclusions

“I have seen the future; it computes.”

It computes geometrically.

Qin08, H. Qin and X. Guan, Physical Review Letters 100, 035006 (2008).

Qin09, H. Qin, X. Guan, W. M. Tang, Physics of Plasmas 16, 042510 (2009).

Li11, J Li, H. Qin, Z. Pu, et al., Phys. Plasmas 18, 052902 (2011).

Squire12, J. Squire, H. Qin, W. M. Tang, Phys. Plasmas 19, 052501 (2012).

Kraus13, “Variational Integrators in Plasma Physics”, Michael Kraus, arXiv:1307.5665 (2015). Zhang14, R. Zhang, J. Liu, Y. Tang, et al., Phys. Plasmas 21, 032504 (2014).

Ellison15, C. L. Ellison, J. M. Finn, H. Qin, W. M. Tang, Plasma Physics and Controlled fusion 57, 054007 (2015).

Qin13, H. Qin, S. Zhang, J. Xiao, et al., Phys. Plasmas 20, 084503 (2013).

Webb14, S. D. Webb, Journal of Computational Physics 270, 570 (2014). He15, Y. He, Y. Sun, J. Liu, H. Qin, Journal of Computational Physics 281, 135 (2015).

ZhangS15, S. Zhang, Y. Jia, Q. Sun, Journal of Computational Physics 282, 43 (2015).

Ellison15c, “Comment on 'Symplectic Integration of Magnetic Systems’”, C. L. Ellison, J.W. Burby, H. Qin, submitted, (2015).

Zhang15, R. Zhang, J. Liu, H. Qin, et al., Physics of Plasmas 22, 044501 (2015)

Squire12b, J. Squire, H. Qin, W. M. Tang, Phys. Plasmas 19, 084501 (2012). Xiao13, J. Xiao, J. Liu , H. Qin, et al., Phys. Plasmas 20, 102517 (2013).

Evstatiev13, E.G. Evstatiev and B.A. Shadwick, Journal of Computational Physics 245, 376 (2013).

Evstatiev14, E.G. Evstatiev , Computer Physics Communications 185, 2851 (2013).

References

Qin15, “Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov-Maxwell system”, H. Qin, J. Liu, J. Xiao, et al., submitted, arXiv:1503.08334 (2015).

Crouseilles15, N. Crouseilles, L. Einkemmer, E. Faou, Journal of Computational Physics 283, 224 (2015).

Qin15b, H. Qin, Y. He, R. Zhang, et al., Journal of Computational Physics 297, 721 (2015).

He15b, “Hamiltonian integration methods for Vlasov-Maxwell equations”, Y. He, H. Qin, Y. Sun, et al., arXiv:1505.06076 (2015).

Burby15, J.W. Burby, A. J. Brizard, P.J. Morrison, H. Qin, Physics Letter A, in press (2015), arXiv:1411.1790v1.

Qin07, H. Qin, R. H. Cohen, W. M. Nevins, X. Xu, Physics of Plasmas 14, 056110 (2007).

Zhou14, Y. Zhou, H. Qin, J. W. Burby, A. Bhattacharjee, Phys. Plasmas 21, 102109 (2014).

References