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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 |
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.
References:
[1] |
P. M. Bellan,
Fundamentals of Plasma Physics, 1$^{st}$ edition, Cambridge University Press, 2008.
doi: 10.1017/CBO9780511807183. |
[2] |
C. K. Birdsall and A. B. Langdon,
Plasma Physics via Computer Simulation, Ser. Plasma Phys. Taylor & Francis, 2005. |
[3] |
J. Boris,
Proceedings of the fourth conference on numerical simulation of plasmas, Washington D.C., (1970), 3-67.
|
[4] |
S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories,
Phy. Rev. E, 77 (2008), 066401, 12pp. |
[5] |
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. |
[6] |
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. |
[7] |
K. Feng and Z. Shang,
Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.
doi: 10.1007/s002110050153. |
[8] |
E. Hairer, C. Lubich and G. Wanner,
Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006. |
[9] |
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. |
[10] |
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. |
[11] |
R. G. Littlejohn,
Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749.
doi: 10.1063/1.863594. |
[12] |
R. G. Littlejohn,
Variational principles of guiding centre motion, J. Plasma Phys., 29 (1983), 111-125.
doi: 10.1017/S002237780000060X. |
[13] |
B. Leimkuhler and S. Reich,
Simulating Hamiltonian Dynamics, Cambridge: Cambridge University Press, 2004. |
[14] |
J. E. Marsden and T. S. Ratiu,
Introduction to Mechanics and Symmetry, Springer-Verlag, Berlin, 1999. |
[15] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[16] |
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. |
[17] |
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. |
[18] |
Z. Shang,
Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.
|
[19] |
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. |
[20] |
Y. Sun,
A class of volume-preserving numerical algorithms, Appl. Math. Comput., 206 (2008), 841-852.
doi: 10.1016/j.amc.2008.10.004. |
[21] |
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. |
[22] |
S. D. Webb,
Symplectic intergration of magnetic systems, J. Comput. Phys., 270 (2014), 570-576.
doi: 10.1016/j.jcp.2014.03.049. |
[23] |
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. |
[24] |
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. |
show all references
References:
[1] |
P. M. Bellan,
Fundamentals of Plasma Physics, 1$^{st}$ edition, Cambridge University Press, 2008.
doi: 10.1017/CBO9780511807183. |
[2] |
C. K. Birdsall and A. B. Langdon,
Plasma Physics via Computer Simulation, Ser. Plasma Phys. Taylor & Francis, 2005. |
[3] |
J. Boris,
Proceedings of the fourth conference on numerical simulation of plasmas, Washington D.C., (1970), 3-67.
|
[4] |
S. A. Chin, Symplectic and energy-conserving algorithms for solving magnetic field trajectories,
Phy. Rev. E, 77 (2008), 066401, 12pp. |
[5] |
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. |
[6] |
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. |
[7] |
K. Feng and Z. Shang,
Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.
doi: 10.1007/s002110050153. |
[8] |
E. Hairer, C. Lubich and G. Wanner,
Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006. |
[9] |
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. |
[10] |
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. |
[11] |
R. G. Littlejohn,
Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749.
doi: 10.1063/1.863594. |
[12] |
R. G. Littlejohn,
Variational principles of guiding centre motion, J. Plasma Phys., 29 (1983), 111-125.
doi: 10.1017/S002237780000060X. |
[13] |
B. Leimkuhler and S. Reich,
Simulating Hamiltonian Dynamics, Cambridge: Cambridge University Press, 2004. |
[14] |
J. E. Marsden and T. S. Ratiu,
Introduction to Mechanics and Symmetry, Springer-Verlag, Berlin, 1999. |
[15] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[16] |
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. |
[17] |
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. |
[18] |
Z. Shang,
Construction of volume-preserving difference schemes for source-free systems via generating functions, J. Comput. Math., 12 (1994), 265-272.
|
[19] |
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. |
[20] |
Y. Sun,
A class of volume-preserving numerical algorithms, Appl. Math. Comput., 206 (2008), 841-852.
doi: 10.1016/j.amc.2008.10.004. |
[21] |
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. |
[22] |
S. D. Webb,
Symplectic intergration of magnetic systems, J. Comput. Phys., 270 (2014), 570-576.
doi: 10.1016/j.jcp.2014.03.049. |
[23] |
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. |
[24] |
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. |







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