Abstract
A test problem for the ‘particle-in-cell’ method is proposed which allows one to check individual errors on each stage of numerical implementation of the method. General testing usually controls only total (final) error. Using the test problem, we analyze errors of a difference method of MacCormack type and the CIC method being the most popular version of the ‘particle-in-cell’ method. It is shown that the CIC method having also the second formal order of accuracy is seriously inferior to the difference method under the same given error and same grid parameters. In particular, the results of the particle method lose their sense after approximately one calculation period, although the time interval of confidence in simulation data for the difference method is practically unlimited. The testing method proposed here is suitable for an arbitrary one-dimensional version of the ‘particle-in-cell’ method, and it allows one not only to compare numerical implementations of individual stages with each other, but also to verify available theoretical results.
References
[1] A. F. Aleksandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics. Springer, New York, 1984.10.1007/978-3-642-69247-5Search in Google Scholar
[2] N. E. Andreev, L. M. Gorbunov, and R. R. Ramazashvili, Theory of a three-dimensional plasma wave excited by a high-intensity laser pulse in an underdense plasma. Plasma Physics Reports 23 (1997) No. 4, 277–284.Search in Google Scholar
[3] N. S. Bakhvalov, A. A. Kornev, and E. V. Chizhonkov, Numerical Methods: Solutions of Problems and Exercises, 2nd ed. Laboratoriya Znanii, Moscow, 2016 (in Russian).Search in Google Scholar
[4] C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation. Series in Plasma Physics and Fluid Dynamics. CRC Press, Boca Raton, 2004.Search in Google Scholar
[5] E. V. Chizhonkov, Mathematical Aspects of Modelling Oscillations and Wake Wavesin Plasma. CRC Press, Boca Raton, 2019.10.1201/9780429288289Search in Google Scholar
[6] E. V. Chizhonkov, On second-order accuracy schemes for modeling of plasma oscillations. Numerical Methods and Programming 21 (2020), 115–128 (in Russian).10.26089/NumMet.v21r110Search in Google Scholar
[7] E. V. Chizhonkov, Rusanov’s third-order accurate scheme for modeling plasma oscillations. Comp. Math. Math. Phys. 63 (2023), No. 5, 905–918.10.1134/S096554252305007XSearch in Google Scholar
[8] G.-H. Cottet and P.-A. Raviart, Particle methods for one-dimensional Vlasov–Poisson equations. SIAM J. Numer. Anal. 21 (1984), No. 1, 52–76.10.1137/0721003Search in Google Scholar
[9] R. C. Davidson, Methods in Nonlinear Plasma Theory. Acad. Press, New York, 1972.Search in Google Scholar
[10] J. M. Dawson, Nonlinear electron oscillations in a cold plasma. Phys. Rev. 113 (1959), No. 2, 383–387.10.1103/PhysRev.113.383Search in Google Scholar
[11] G. Dimarco and L. Pareschi, Numerical methods for kinetic equations. Acta Numerica 23 (2014), 369–520.10.1017/S0962492914000063Search in Google Scholar
[12] A. A. Frolov and E. V. Chizhonkov, On numerical simulation of travelling Langmuir waves in a warm plasma. Matematicheskoe Modelirovanie 35 (2023), No. 11, 21–34.10.20948/mm-2023-11-02Search in Google Scholar
[13] Yu. N. Grigor’ev, V. A. Vshivkov, and M. P. Fedoruk, Numerical Simulation by Particle-in-Cell Methods. Publ. SB RAS, Novosibirsk, 2004 (in Russian).Search in Google Scholar
[14] L. M. Gorbunov, Why do we need super-powerful laser pulses? Priroda 21(2007), No. 4, 11–20.Search in Google Scholar
[15] R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles. McGraw-Hill Inc., New York, 1981.Search in Google Scholar
[16] R. W. MacCormack, The effect of viscosity in hypervelocity impact cratering. J. Spacecr. Rockets 40 (2003), No. 5, 757–763.10.2514/2.6901Search in Google Scholar
[17] P. A. Mora and T. M. Antonsen, Kinetic modeling of intense, short pulses propagating in tenuous plasmas. Phys. Plasmas 4 (1997), No. 4, 217–229.10.1063/1.872134Search in Google Scholar
[18] A. Myers, P. Colella, and B. Van Straalen, A 4th-order particle-in-cell method with phase-space remapping for the Vlasov– Poisson equation. SIAM J. Sci. Comput. 39 (2017), No. 3, 467–485.10.1137/16M105962XSearch in Google Scholar
[19] H. Qin, J. Liu, J. Xiao, et al. Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations. Nuclear Fusion 2016. 56 (2016), No. 1, 014001.10.1088/0029-5515/56/1/014001Search in Google Scholar
[20] O. S. Rozanova and E. V. Chizhonkov, On the existence of a global solution of a hyperbolic problem. Doklady Math. 101 (2020), No. 3, 254–256.10.1134/S1064562420030163Search in Google Scholar
[21] O. S. Rozanova and E. V. Chizhonkov, On the conditions for the breaking of oscillations in a cold plasma. Z. Angew. Math. Phys. 72 (2021), No. 1, 13.10.1007/s00033-020-01440-3Search in Google Scholar
[22] O. S. Rozanova and E. V. Chizhonkov, Analytical and numerical solutions of one-dimensional cold plasma equations. Comp. Math. Math. Phys. 61 (2021), No. 9, 1485–1503.10.1134/S0965542521090141Search in Google Scholar
[23] M. H. Schultz, Spline Analysis. Prentice-Hall Int., New York, 1973.Search in Google Scholar
[24] C. J. R. Sheppard, Cylindrical lenses—focusing and imaging: a review. Applied Optics 52 (2013), No. 4, 538–545.10.1364/AO.52.000538Search in Google Scholar PubMed
[25] V. P. Silin, Introduction to Kinetic Theory of Gases. Nauka, Moscow, 1971 (in Russian).Search in Google Scholar
[26] L. Tonks and I. Langmuir, Oscillations in ionized gases. Phys. Rev. 33 (1929), No. 2, 195–210.10.1103/PhysRev.33.195Search in Google Scholar
[27] A. A. Vlasov, The vibrational properties of an electron gas. Sov. Phys. Usp. 10 (1968), 721–733.10.1070/PU1968v010n06ABEH003709Search in Google Scholar
[28] B. Wang, G. H. Miller, and P. Colella, A particle-in-cell method with adaptive phase-space remapping for kinetic plasmas. SIAM J. Sci. Comp. 33 (2011), No. 6, 3509–3537.10.1137/100811805Search in Google Scholar
[29] R. P. Wilhelm and M. Kirchhart, An interpolating particle method for the Vlasov–Poisson equation. J. Comp. Phys. 473 (2023), 111720.10.1016/j.jcp.2022.111720Search in Google Scholar
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