Variational Symplectic Integrator of the Guiding Center...

Hong Qin1,2, Xiaoyin

Guan1, William M. Tang1

Princeton Plasma Physics Laboratory, Princeton UniversityCenter for Magnetic Fusion Theory, Chinese Academy of Sciences

US-Japan Workshop on Progress of Multi-Scale Simulation ModelsNov. 20, 2008, Dallas, USA

www.pppl.gov/~hongqin/Gyrokinetics.php

Variational

Symplectic

Integrator

of the Guiding Center Motion






How to Calculate the

Guiding Center Motion?

You Don’t How to Calculate

the Guiding Center Motion!






Gyrocenter

dynamics and algorithms

( )2u bd Bu

dt B B Bdu Bdt B

m

m

´ ⋅´ ´= + + +

⋅=- + ⋅

b bX b E bb

bb E

nth (4th) order Runge-Kutta

methods

Long time non-conservation.

Errors add up coherently.

nth (4th) order Runge-Kutta

methods

Long time non-conservation.

Errors add up coherently.

Carl Runge

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

Example –

banana orbit go bananas

Numerical result by RK4

Exact banana orbit

49 10 turns´

~

~

6 burn banana

6run banana

ITER: 3 10

EAST: 10 10

T T

T T

´

´

H. Qin and X. Guan, PRL 100, 035006 (2008). X. Xiao and S. Wang, submitted to PoP.

/ 100t TD =

Example –

passing orbit go bananas too

49 10 turns´


Exact passing orbit

~

~

6 burn banana

6run banana

ITER: 3 10

EAST: 10 10

T T

T T

´

´

/ 33t TD =

Numerical banana

60 million years ago ……

Can we do better than RK4? --

symplectic

integrator

Feng

(1983) Ruth (1984)

Conserves symplectic

structure;Bound energy error globally

Requires canonical Hamiltonian structure

Symplectic

integrator ?Symplectic

integrator ?

,

,

0 1

1 0q

p

Hq

Hp

æ öæ ö æ ö ÷÷ ÷çç ç= ÷÷ ÷çç ç ÷÷ ÷ç ç ÷çè ø è ø- è ø

Application areas:

Accelerator physics (everybody)

Planetary dynamics (S. Tremaine)

Nonlinear dynamics (everybody)

Plasma physics (J. Cary 89’)

What is the canonical structure for gyrocenter

dynamics?

R. White

Gyrocenter

dynamics does not have a global canonical structure

Only has a non-canonical

symplectic structure dwWº

0i dt g =

Worldline

Inner product

Exterior derivative

Hamilton

( )( )

212u B dt

A u d

g w m f

w

= - + +

= + ⋅b X

Geometry of gyrocenterGeometry of gyrocenter

Gyrocenter

dynamics

0i dt g =

† †

† †

† †

†

( )

,

2

0

du

dt B B

dudt B

d dB

dt dt

m

q m

´= + ⋅ ´ -

⋅=

= =

X B b Eb b

B E

† † †

† † †,

u

B Bm

º ´ , º +

= ⋅ º -

B A A A b

B b E E

» R. Littlejohn( ) ( )21

2 L u u Bmq m f= + ⋅ + - + +A b X

, driftB´ E B

curvaturedrift

( ) ( )212

u d d u B dtg m q m f= + ⋅ + - + +A b X

Banos

drift (1967)

E-L Eq.

演示者

演示文稿备注

After finding the new, good gyrocenter coord, we find the new gamma. It is not really the new gamma. It is the old gamma in the new coord. Physics never changes, just you look it from a different angle. The gamma description is equivalent to the Lagrangian L. This Lagrangian was first derived by Littlejohn when he was here at UCLA. The guiding center dynamics you get either from the Hamiltonian eq. or the E-L Eq. is this. Here is your EXB drift and grad-B drift through the definition of E^dagger. The curvature drift is hidden inside the definition of B^dagger. I would like to bring your attention to this term called Banos drift. It is an extra drift in the parallel direction, in addition to u, found by Alfredo Banos in 1967 here in UCLA. Very very few people appreciate the value of this term. But it is as important as the curvature drift. It is much more difficult to find this drift because it is in the parallel direction. It escaped from the eyes of all the first generation of plasma physicist, except for Banos. You have to be very precise and careful to find this drift. For this, he has my admiration. The official history of your dept. says that “Banos was a precise man, an impeccable dresser and had a flair for the dramatic”, which is clear from this picture. Why is it important? If you throw it away. The Phase space volume is not conserved. Louvilles’ Theorem is violated. Of course, Banos found this term by very precise elementary analysis, not based on the phase space volume conservation. Now we find it as a natural outcome from the geometric rendition of the guiding center dynamics, which automatically gurantee the phase space volume conservation. Alfredo Banos Jr., arrived in the department in 1946 and occupied a second-floor office in Kinsey Hall. He was one of the first physicists to pursue theoretical plasma physics. Years later, after Knudsen Hall was completed, a new second-story bridge was planned to connect the two buildings. Unfortunately, the bridge completely eliminated the office that Baños had so carefully occupied. To compensate, his friends in the department named the connecting footway Baños Bridge. It was formally dedicated in 1963.

Darboux

Theorem

Jean Gaston Darboux(1842-1917)

Darboux

Theorem (1882):

Every symplectic

structure is locally

canonical.

Darboux

Theorem (1882):

Every symplectic

structure is locally

canonical.

Gyrocenter

dynamics can be canonical locally.

No symplectic

integrator for gyrocenter?

Variational

symplectic

integrator

( ) ( )

1

0

212

tA Ldt

L u u Bm f

=

= + ⋅ - + +

òA b X

[ ..., ( ) ]discretize on

0 2 1t h h N h= , , , -

J. Marsden (2001)

( )

( ) ( )

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

-

=

+ +

» = , +

, + º , , ,

åx x

0i dt g = »

( ) ( )[ ]

( ) ( )[ ]

( )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 kux

discretized

Euler- Lagrangian

Eq.

1 1

1 1

,k k k k

k k

u u

u

x x

x

Conserved symplectic

structure

( ) ( )

( ) ( )

( )

( )

1 11 1

1 1 1

1 1 1

d k d kk k

d k d kk k

k k L k k d L k k duu

k k L k k d L k k duu

q

q

++ +

+ +

-

¶ ¶, + º , + ⋅ + , +

¶ ¶

¶ ¶, + º- , + ⋅ - , +

¶ ¶

xx

xx

( ) ( ) ( )1 1 1ddL k k k k k kq q+ -, + = , + - , +

( )1d k k d dq q+ -W , + º = ( ) ( )0 1 1ddA N Nq q+ -= , - - ,

( ) ( )0 1 1d d N NW , =W - ,

minimize w.r.t. k kux

1st

order variational

symplectic

integrator

( )( ) ( )

( ) ( )† †

1 111

2 2k k k k

d

k k u uL k k B k k

hm j

é ùê ú+ +ë ûé ù+ + -ë û, + º ⋅ - - -A A x x

( ) ( ) ( )

( )( )

† † †( ) ( ) ( )1 1

1 11 1

1 11 1

2 212 2

i i i j jk k j jj

k ki i ik k

A k x x A k A k B k kh h

u ub k x xh

m j+ - , ,,

+ -+ -

é ù- - + - - = +ë û

+- =

implicitimplicit

1 1

1 1

,k k k k

k k

u u

u

x x

x

H. Qin and X. Guan, PRL 100, 035006 (2008).

Semi-explicit Newton’s method

( ) ( )( )†† †( ) ( ) ( ) 1 1 1 11 1 jj j i i jk k k kiA k A k A k x x b k u u+ - + -,

é ù+ - - » - + -ë û

( ) ( ) ( )( )

( ) ( )

( )( )

† †( ) ( ) 1 1 1 1 1

1 11 1

12

2 2

12 2

j ii j i i i i

k k k k kj i

j j

k ki i ik k

b k b kA k A k x x u x x

h h h

B k k

u ub k x xh

m j

é ùê úê ú+ - - + -, , ê úê úë û

, ,

+ -+ -

é ù- - - - -ë û

= +

+- = explicit, initial guess for

the Newton’s method

explicit, initial guess for the Newton’s method

Variational

symplectic

integrator, banana orbits

/ 33t TD =/ 100t TD =

49 10 turns´Transport reduction by integration errors

Variationalsymplectic


RK4RK4

H. Qin and X. Guan, PRL 100, 035006 (2008).

Variational

symplectic

integrator bounds energy error globally

Parabolic growth of energy error

burn0.03 T

Variational

symplectic

integrator, no numerical banana

/ 100t TD =

49 10 turns´Transport reduction by integration errors


VariationalsymplecticRK4RK4 / 33t TD =

Numerical banana

Variational

symplectic

integrator bounds energy error globally


. burn0 03 T

Conclusions

Physics is geometry. So is algorithm.Physics is geometry. So is algorithm.

Symplectic

bounds error globally, others do not.Symplectic

bounds error globally, others do not.

How to calculate guiding center motion?How to calculate guiding center motion?

Multi-scale dynamics needs global algorithms

How about collisions?

Collisions do not remove the energy errors of RK4.Collisions do not remove the energy errors of RK4.

Global energy errors of RK4 are numerical noises to collisional

physics.

Global energy errors of RK4 are numerical noises to collisional

physics.

Frequent heartburns do not cure stomach cancer.Frequent heartburns do not cure stomach cancer.

How about implicitness?

Implicit root search does destroy the symplectic

structure.Implicit root search does destroy the symplectic

structure.

Approximate solution is much better than no solution.Approximate solution is much better than no solution.


VariationalsymplecticRK4RK4

Example –

gradient drift

22Exact gyrocenter orbit 1

4x

y+ =

( )

( )

ˆ2

21 0 054

B x y

xB x y y

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

= ,

, = + . +

B z

55 10 turns´

Example 1 –

gradient drift

22

Exact gyrocenter

orbit 14x

y+ =

( )2

21 0 054zx

B x y yæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

, = + . +

55 10 turns´


Variational

symplectic

globally bounds energy error

Energy error (Orbit deviation)

Energy error (Orbit deviation)


Variational Symplectic Integrator of the Guiding Center...

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