Variational Symplectic Integrator of the Guiding Center...
Transcript of 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
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
How to Calculate the
Guiding Center Motion?
You Don’t How to Calculate
the Guiding Center Motion!
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
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´
Numerical result by RK4
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.
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
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
Variationalsymplectic
VariationalsymplecticRK4RK4 / 33t TD =
Numerical banana
Variational
symplectic
integrator bounds energy error globally
Parabolic growth of energy error
. 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.
Variationalsymplectic
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´
Numerical result by RK4
Variational
symplectic
globally bounds energy error
Energy error (Orbit deviation)
Energy error (Orbit deviation)
Parabolic growth of energy error