1. No category

Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Volume-Preserving Algorithm for Secular
Relativistic Dynamics of Charged Particles
Ruili Zhang,
Joint work with Jian Liu, Hong Qin, Yulei Wang, Yang He and Yajuan Sun
Department of Modern Physics
University of Science and Technology of China
rlzhang@ustc.edu.cn
Mar 24th, 2015
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Outline
1
Secular relativistic dynamics
2
Systematic method to construct VPAs
3
Numerical experiments
4
Summary
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Dynamics of relativistic charged particles serves as the first principle model
underlying plasma physics, accelerator physics, astrophysics, space physics,
and many other subfields of physics!
The long-term relativistic dynamics is critical to describe various multi-timescale
problems, such as runaway electrons in tokamaks.
Ruili Zhang et al.
VPA for relativistic charged particles
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
The relativistic charged particle dynamics is
dx
p
= p 2
,
dt
m0 + p2 /c2
(1)
dp
p
× B(x, t)
= q E(x, t) + p 2
dt
m0 + p2 /c2
!
.
x and p denote position and momentum vector
Ruili Zhang et al.
VPA for relativistic charged particles
(2)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Difficulties:
Time-dependent electromagnetic fields
How to deal with the relativistic factor?
Multi-scale problem
Long-time simulation
Geometric numerical integrator!
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Geometric Properties:
6-form is an invariant
Liouville’s theorem
The solution flow ϕt : z(t0 ) → z(t) preserves the phase space volume V(t), i.e.,
Volume-preserving algorithm (VPA)!
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Volume-preserving algorithm:
Φh (xk , pk ) = (xk+1 , pk+1 )
Is there a systematic method to construct VPAs?
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
This VPA is constructed using the splitting method consisting of three steps:
(1) To split the original system into several incompressible subsystems.
(2) To find a VPA for each subsystem.
(3) To combine the sub-algorithms into a desired VPA for the original
system.
Ruili Zhang et al.
VPA for relativistic charged particles
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Expressing using Lie-derivative
Correspondence between Lie algebras and Lie groups
BCH formula
exp(tXF ) = exp(tXF1 ) exp(tXF2 ) + O(t) ,
t
t
exp(tXF ) = exp( XF1 ) exp(tXF2 ) exp( XF1 ) + O(t2 ) ,
2
2
t
t
t
t
exp(tXF ) = exp( XF1 ) exp( XF2 ) exp(tXF3 ) exp( XF2 ) exp( XF1 ) + O(t2 ) .
2
2
2
2
Ruili Zhang et al.
VPA for relativistic charged particles
(3)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Step 1: To split the original system
Step 1: To split the original system into three subsystems.


dx
p


,
= p 2

 dx = 0 ,
m0 + p2 /c2
dt
S1 := dt
S2 := dp
 dp



= qE(x, t) ,
=0,
dt
dt

dx


=0,

dt
!
S3 := dp
p


× B(x, t) .
 dt = q p 2
m0 + p2 /c2
Each subsystem is source-free system!
Ruili Zhang et al.
VPA for relativistic charged particles
(4)
(5)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Step 2: To find a VPA method for each subsystem.
Step 2: To find a VPA method for each subsystem.
The exact solution of S1 :

xk+1 = xk + ∆t p pk
,
1
m20 + p2k /c2
ϕ (∆t) :

pk+1 = pk .
Symmetric VPA of order 2 for subsystems S2 using mid-point rule:

xk+1 = xk ,
2
ϕ (∆t) :
pk+1 = pk + q∆tE(xk , tk + ∆t ) ,
2
Time-dependent electromagnetic fields. X
Ruili Zhang et al.
VPA for relativistic charged particles
(6)
(7)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
For S3 , p2 (t) = p20 . The relativistic factor is an invariant γ =
Summary
p
m20 + p20 /c2
S3 is equavalent to

dx

=0,

dt
S3 := dp
q

= p 2
(p × B(x, t)) .

dt
m0 + p20 /c2
0
(8)
Symmetric VPA of order 2 for subsystems S3 using mid-point rule:

x
= xk ,
(9a)

 k+1
!
3
ϕ (∆t) :
pk + pk+1
∆t

p
× B(xk , tk +
) (.9b)
 pk+1 = pk + q∆t
2
2
2
2
2 m0 + pk /c
How to deal with the relativistic factor? X
Ruili Zhang et al.
VPA for relativistic charged particles
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Step 3: Composition the sub-VPAs
Step 3:Composition the sub-VPAs
Φ(∆t) = ϕ1 (
∆t
∆t
∆t
∆t
) ◦ ϕ2 ( ) ◦ ϕ3 (∆t) ◦ ϕ2 ( ) ◦ ϕ1 ( ) .
2
2
2
2
The resulting VPA using the combination in Eq. (10) is

∆t
pk


,
xk+ 12 = xk + 2 p 2

m
+
p2k /c2

0



∆t


p− = pk + q Ek+ 1 ,


2
2






ˆ k+1/2
 +
q∆tB
 q
 p− ,
Φ(∆t) : p = Cay
2
2
−
2

2 m0 + p /c




∆t

+

pk+1 = p + q Ek+ 1 ,


2

2


∆t
pk+1


q
x
=
x
+
.
1
k+1

k+ 2

2
2

m0 + p2k+1 /c2
Ruili Zhang et al.
VPA for relativistic charged particles
(10)
(11)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Cayley transform:
−1 ˆ 1 p− = Cay(aB
ˆ 1 )p− ,
ˆ 1
I + aB
p+ = I − aB
k+
k+
k+
2
2
2
Summary
(12)
ˆ 1 is an anti-symmetric matrix, and
where B
k+
2
q∆t
.
a= p 2
2 m0 + (p− )2 /c2
(13)
It can be manipulated explicitly.
p+ =
I+
2a
2a2
ˆ 1 +
ˆ2 1
B
B
k+ 2
2
2
1 + a |Bk+ 1 |
1 + a2 |Bk+ 1 |2 k+ 2
2
Ruili Zhang et al.
!
p− .
2
VPA for relativistic charged particles
(14)
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Advantages:
It preserve phase space volume exactly!
It is symmetric!
It can be computed explicitly!
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Numerical experiments
Multi-scale problem:
the runaway electron runs at the speed nearly 3 × 108 m/s;
the outward neoclassical drift velocity only about 3m/s.
B0
B0 R0
B=−
eζ −
R
R0
E = − Et eζ ,
R
p
(R − R0 )2 + z2
eθ ,
2R
(16)
where B0 = 3T, R0 = 1.7m and Et = 3V/m.
Ruili Zhang et al.
(15)
VPA for relativistic charged particles
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Figure: The long-term simulation results of runaway electron dynamics in tokamak
running 6.28 × 1010 time steps by VPA.
Ruili Zhang et al.
VPA for relativistic charged particles
Summary
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Figure: The simulation result by the Boris algorithm, as a comparison to the VPA. (a)
The snapshots of runaway drift orbits calculated by the Boris algorithm at t=0s,
t=10−4 s, and t=1.07 × 10−4 s.
Multi-scale problem X
Long-time simulation X
Ruili Zhang et al.
VPA for relativistic charged particles
Secular relativistic dynamics
Systematic method to construct VPAs
Numerical experiments
Summary
Summary
A systematic method to construct VPAs for relativistic charged particles
dynamics has been given.
Relative error of energy can be bounded.
Relative error of canonical angular momemtum can be bounded.
The VPA enables the discovery of new physics for secular runaway
dynamics.
High order VPAs can be obtained using the splitting method.
Thank you for you attention!
Ruili Zhang et al.
VPA for relativistic charged particles