DOI: https://doi.org/10.26089/NumMet.v25r105

On errors in the PIC-method when modeling Langmuir oscillations

Authors

  • Evgenii V. Chizhonkov

Keywords:

collisionless plasma
nonlinear Langmuir oscillations
analytical solution
numerical modeling
PIC-method
difference method
error analysis

Abstract

A test problem is constructed that simulates nonlinear Langmuir oscillations excited by a short powerful laser pulse. The problem has an analytical solution in Lagrangian coordinates, which can be transformed into Eulerian coordinates, however, in a specific implicit form. For various variants of the particle-in-cell method (PIC-method), analytical estimates of the error that occurs when the assembly of the charge from the macroparticles into the centers of the cells. In addition, numerical experiments have been carried out to illustrate the quality of these estimates on a model problem. Additionally, a new difference method for the model problem is proposed and its accuracy is compared with the charge assembly stage from the PIC-method.

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Published

2024-02-16

Issue

Vol. 25 (2024): Issue 1.

Section

Methods and algorithms of computational mathematics and their applications

Author Biography

Evgenii V. Chizhonkov

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• Professor

References

  1. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Springer, Cham, 2016).
    https://doi.org/10.1007/978-3-319-22309-4 . Cited February 3, 2024.
  2. A. A. Vlasov, “The Vibrational Properties of an Electron Gas,” Usp. Fiz. Nauk 93 (3), 444-470 (1967) [Sov. Phys. Usp. 10 (6), 721-733 (1968)].
    doi 10.1070/PU1968v010n06ABEH003709
  3. T. D. Arber and R. G. L. Vann, “A Critical Comparison of Eulerian-Grid-Based Vlasov Solvers,” J. Comput. Phys. 180 (1), 339-357 (2002).
    doi 10.1006/jcph.2002.7098
  4. F. Filbet and E. Sonnendrücker, “Comparison of Eulerian Vlasov Solvers,” Comput. Phys. Commun. 150 (3), 247-266 (2003).
    doi 10.1016/S0010-4655(02)00694-X
  5. G. Dimarco and L. Pareschi, “Numerical Methods for Kinetic Equations,” Acta Numerica 23, 369-520 (2014).
    doi 10.1017/S0962492914000063
  6. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, New York, 1981).
  7. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation. In Series in Plasma Physics and Fluid Dynamics (CRC Press, Boca Raton, 2004).
  8. Yu. N. Grigor’ev, V. A. Vshivkov, and M. P. Fedoruk, Numerical Simulation by Particle-in-Cell Methods (Ross. Akad. Nauk, Novosibirsk, 2004) [in Russian].
  9. L. Tonks and I. Langmuir, “Oscillations in Ionized Gases,” Phys. Rev. 33 (2), 195-210 (1929).
    doi 10.1103/PhysRev.33.195
  10. G.-H. Cottet and P.-A. Raviart, “Particle Methods for the One-Dimensional Vlasov-Poisson Equations,” SIAM J. Numer. Anal. 21 (1), 52-76 (1984).
    doi 10.1137/0721003
  11. B. Wang, G. H. Miller, and P. Colella, “A Particle-in-Cell Method with Adaptive Phase-space Remapping for Kinetic Plasmas,” SIAM J. Sci. Comput. 33 (6), 3509-3537 (2011).
    doi 10.1137/100811805
  12. V. A. Vshivkov, “The Approximation Properties of the Particles-in-Cells Method,” Zh. Vychisl. Mat. Mat. Fiz. 36 (4), 106-113 (1996) [Comput. Math. Math. Phys. 36 (4), 509-515 (1996)].
  13. A. F. Alexandrov, L. S. Bogdankevich, and A. A. Rukhadze, Principles of Plasma Electrodynamics (Vysshaya Shkola, Moscow, 1978; Springer, Berlin, 1984).
    https://link.springer.com/book/9783642692499 . Cited February 4, 2024.
  14. V. L. Ginzburg and A. A. Rukhadze, Waves in Magnetoactive Plasma (Nauka, Moscow, 1975) [in Russian].
  15. R. C. Davidson, Methods in Nonlinear Plasma Theory (Academic Press, New York, 1972).
  16. E. V. Chizhonkov, Mathematical Aspects of Modelling Oscillations and Wake Waves in Plasma (Fizmatlit, Moscow, 2018; CRC Press, Boca Raton, 2019).
  17. O. S. Rozanova and E. V. Chizhonkov, “Analytical and Numerical Solutions of One-Dimensional Cold Plasma Equations,” Zh. Vychisl. Mat. Mat. Fiz. 61 (9), 1508-1527 (2021). [Comput. Math. Math. Phys. 61 (9), 1485-1503 (2021)].
    doi 10.1134/S0965542521090141
  18. A. A. Frolov and E. V. Chizhonkov, “On Numerical Simulation of Travelling Langmuir Waves in a Warm Plasma,” Mat. Model. 35 (11), 21-34 (2023).
    doi 10.20948/mm-2023-11-02
  19. L. M. Gorbunov, “What Superpower Laser Pulses Are Needed for?’’ Priroda 21 (4), 11-20 (2007).
    https://priroda.ras.ru/pdf/2007-04.pdf . Cited February 5, 2024.
  20. P. A. Mora and T. M. Antonsen, “Kinetic Modeling of Intense, Short Laser Pulses Propagating in Tenuous Plasmas,” Phys. Plasmas 4 (1), 217-229 (1997).
    doi 10.1063/1.872134
  21. 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 (4), 277-284 (1997).
  22. C. J. R. Sheppard, “Cylindrical Lenses -- Focusing and Imaging: a Review [Invited],” Appl. Opt. 52 (4), 538-545 (2013).
    doi 10.1364/AO.52.000538
  23. O. S. Rozanova and E. V. Chizhonkov, “On the Existence of a Global Solution of a Hyperbolic Problem,” Dokl. Akad. Nauk 492 (1), 97-100 (2020) [Dokl. Math. 101 (3), 254-256 (2020)].
    doi 10.1134/S1064562420030163
  24. O. S. Rozanova and E. V. Chizhonkov, “On the Conditions for the Breaking of Oscillations in a Cold Plasma,” Z. Angew. Math. Phys. 72 (1), Article Number 13 (2021).
    doi 10.1007/s00033-020-01440-3
  25. E. V. Chizhonkov, “Rusanov’s Third-Order Accurate Scheme for Modeling Plasma Oscillations,” Zh. Vychisl. Mat. Mat. Fiz. 63 (5), 864-878 (2023) [Comput. Math. Math. Phys. 63 (5), 905-918 (2023)].
    doi 10.1134/S096554252305007X
  26. J. M. Dawson, “Nonlinear Electron Oscillations in a Cold Plasma,” Phys. Rev. 113 (2), 383-387 (1959).
    doi 10.1103/PhysRev.113.383
  27. 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].
  28. R. W. MacCormack, “A Numerical Method for Solving the Equations of Compressible Viscous Flow,” AIAA J. 20 (9), 1275-1281 (1982).
  29. E. V. Chizhonkov, “On Second-Order Accuracy Schemes for Modeling of Plasma Oscillations,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 21 (1), 115-128 (2020).
    doi 10.26089/NumMet.v21r110
  30. Yu. I. Shokin and N. N. Yanenko, The Method of Differential Approximation. Application to Gas Dynamics (Nauka, Novosibirsk, 1985) [in Russian].
  31. J. Fürst and P. Furmánek, “An Implicit MacCormack Scheme for Unsteady Flow Calculations,” Comput. Fluids 46 (1), 231-236 (2011).
    doi 10.1016/j.compfluid.2010.09.036
  32. 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,” Nucl. Fusion 56 (1), Article Number 014001 (2016).
    doi 10.1088/0029-5515/56/1/014001
  33. 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 (3), B467-B485 (2017).
    doi 10.1137/16M105962X
  34. R. P. Wilhelm and M. Kirchhart, “An Interpolating Particle Method for the Vlasov-Poisson Equation,” J. Comput. Phys. 473, Article Identifier 111720 (2023).
    doi 10.1016/j.jcp.2022.111720

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