High order Gauss-Seidel schemes for charged particle dynamics

Yuezheng Gong; Jiaquan Gao; Yushun Wang

doi:10.3934/dcdsb.2018034

Article Contents

2018, Volume 23, Issue 2: 573-585. Doi: 10.3934/dcdsb.2018034

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High order Gauss-Seidel schemes for charged particle dynamics

1.
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2.
School of Computer Science and Technology, Nanjing Normal University, Nanjing 210023, China
3.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Author Bio: E-mail address:gongyuezheng@nuaa.edu.cn (Y. Gong); E-mail address:springf12@163.com (J. Gao); E-mail address:wangyushun@njnu.edu.cn (Y. Wang)
* Corresponding author:wangyushun@njnu.edu.cn(Y. Wang)
* Corresponding author:wangyushun@njnu.edu.cn(Y. Wang)

Received: September 30, 2015

Revised: June 30, 2017

Early access: November 2017

Published: June 2018

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Abstract

Gauss-Seidel projection methods are designed for achieving desirable long-term computational efficiency and reliability in micromagnetics simulations. While conventional Gauss-Seidel schemes are explicit, easy to use and furnish a better stability as compared to Euler's method, their order of accuracy is only one. This paper proposes an improved Gauss-Seidel methodology for particle simulations of magnetized plasmas. A novel new class of high order schemes are implemented via composition strategies. The new algorithms acquired are not only explicit and symmetric, but also volume-preserving together with their adjoint schemes. They are highly favorable for long-term computations. The new high order schemes are then utilized for simulating charged particle motions under the Lorentz force. Our experiments indicate a remarkable satisfaction of the energy preservation and angular momentum conservation of the numerical methods in multi-scale plasma dynamics computations.

Keywords:

Mathematics Subject Classification: 60E10, 60J10, 60J27, 60J35.

Citation:

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Figure 1. The fourth order explicit method $RK4$ is applied to the simple 2D dynamics with step size $h = \pi/10$. (a): The orbit in the first $2691$ steps. (b): The orbit after $2.7\times10^{5}$ steps. (c): Energy error $H^{n}-H^{0}$. (d): Angular momentum error $p_{\xi}^{n}-p_{\xi}^{0}$

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Figure 2. Numerical orbits of the symmetric and volume-preserving methods with time step $h = \pi/10$. (a): The orbit after $5\times10^{5}$ steps by the second order method. (b): The orbit after $2.5\times10^{5}$ steps by the fourth order method

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Figure 3. Convergence rates of numerical solutions by the methods $GS_{h}^{2}$, $\tilde{G}_{h}^{2}$, $GS_{h}^{4}$ and $G_{h}^{4}$

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Figure 4. Left: The errors of the energy. Right: The errors of the angular momentum

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Figure 5. Relative errors of the energy $H$ and the angular momentum $p_{\xi}$ as a function of time $t\equiv nh$. The step size is $h = \pi/10$, and the integration time interval is $[0, 10^{5}h]$

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Figure 6. Numerical orbits. (a): Banana orbit by the $RK4$. (b): Transit orbit by the $RK4$. (c): Banana orbit by the volume-preserving methods. (d): Transit orbit by the volume-preserving methods. The step size is $h = \pi/10$, and the integration time interval is $[0, 5\times10^{5}h]$

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Figure 7. Relative errors of the energy $H$ and the angular momentum $p_{\xi}$ as a function of time $t\equiv nh$. The step size is $h = \pi/10$, and the integration time interval is $[0, 5\times10^{5}h]$

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