Article Contents
Article Contents

# High order Gauss-Seidel schemes for charged particle dynamics

• 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.

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

 Citation:

• 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 $p_{\xi}^{n}-p_{\xi}^{0}$

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

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

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

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 $[0, 10^{5}h]$

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 $[0, 5\times10^{5}h]$

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 $[0, 5\times10^{5}h]$

•  P. M. Bellan, Fundamentals of Plasma Physics, 1$^{st}$ edition, Cambridge University Press, 2008. doi: 10.1017/CBO9780511807183. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Ser. Plasma Phys. Taylor & Francis, 2005. J. Boris , Proceedings of the fourth conference on numerical simulation of plasmas, Washington D.C., (1970) , 3-67. S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories, Phy. Rev. E, 77 (2008), 066401, 12pp. W. N. E  and  X. P. Wang , Numerical methods for the Landau-Lifshitz equation, SIAM J. Numer. Anal., 38 (2000) , 1647-1665.  doi: 10.1137/S0036142999352199. K. Feng, Symplectic, contact and volume-preserving algorithms, in: Z. C. Shi, T. Ushijima (Eds. ), Proc. 1st China-Japan Conf. on Computation of Differential Equations and Dynamical Systems, World Scientific, Singapore, 1993, 1–28. K. Feng  and  Z. Shang , Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995) , 451-463.  doi: 10.1007/s002110050153. E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006. Y. He , Y. J. Sun , J. Liu  and  H. Qin , Volume-preserving algorithms for charged particle dynamics, J. Comput. Phys., 281 (2015) , 135-147.  doi: 10.1016/j.jcp.2014.10.032. Y. He , Z. Zhou , Y. Sun , J. Liu  and  H. Qin , Explicit $K$-symplectic algorithms for charged particle dynamics, Phys. Lett. A, 381 (2017) , 568-573.  doi: 10.1016/j.physleta.2016.12.031. R. G. Littlejohn , Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981) , 1730-1749.  doi: 10.1063/1.863594. R. G. Littlejohn , Variational principles of guiding centre motion, J. Plasma Phys., 29 (1983) , 111-125.  doi: 10.1017/S002237780000060X. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge: Cambridge University Press, 2004. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, Berlin, 1999. J. E. Marsden  and  M. West , Discrete mechanics and variational integrators, Acta Numer., 10 (2001) , 357-514.  doi: 10.1017/S096249290100006X. H. Qin, S. X. Zhang, J. Y. Xiao, J. Liu, Y. J. Sun and W. M. Tang, Why is Boris algorithm so good?, Phys. Plasmas, 20 (2013), 084503. doi: 10.1063/1.4818428. H. Qin and X. Guan, Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields, Phys. Rev. Lett. , 100 (2008), 035006. doi: 10.1103/PhysRevLett.100.035006. Z. Shang , Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994) , 265-272. P. H. Stoltz, J. R. Cary, G. Penn and J. Wurtele, Efficiency of a Boris like integration scheme with spatial stepping, Phys. Rev. Spec. Top. Accel. Beams, 5 (2002), 094001. doi: 10.1103/PhysRevSTAB.5.094001. Y. Sun , A class of volume-preserving numerical algorithms, Appl. Math. Comput., 206 (2008) , 841-852.  doi: 10.1016/j.amc.2008.10.004. X. P. Wang , C. J. Garcia-Cervera  and  W. N. E , A Gauss-Seidel Projection Method for Micromagnetics Simulations, J. Comput. Phys., 171 (2001) , 357-372.  doi: 10.1006/jcph.2001.6793. S. D. Webb , Symplectic intergration of magnetic systems, J. Comput. Phys., 270 (2014) , 570-576.  doi: 10.1016/j.jcp.2014.03.049. Y. K. Wu, E. Forest and D. S. Robin, Explicit symplectic integrator for s-dependent static magnetic field, Phys. Rev. E, 68 (2003), 046502. doi: 10.1103/PhysRevE.68.046502. S. X. Zhang , Y. S. Jia  and  Q. Z. Sun , Comment on "Symplectic integration of magnetic systems" by Stephen D. Webb [J. Comput. Phys. 270 (2014) 570-576], J. Comput. Phys., 282 (2015) , 43-46.  doi: 10.1016/j.jcp.2014.10.062.

Figures(7)