Abstract
The numerical simulation of hot and low density plasmas using the Vlasov–Poisson model is necessary for many practical applications such as the characterization of laboratory and astrophysical plasmas. The numerical treatment of the Vlasov equation is addressed using Eulerian methods when high precision and low noise are required. Among these methods, we highlight those based on finite-volumes without splitting, which have shown to be a good option for capturing small structures in phase space with low dissipation while preserving positivity. The problem is that kinetic simulations usually require the discretization of 3–6-dimensional phase-spaces which results in a huge number of ordinary differential equations (ODEs). This stands out the importance of using efficient schemes for the time integration. In this article, linear multistep methods are implemented for the time stepping of the resulting equations, and compared against traditional Runge–Kutta ones. Schemes with built-in error estimation are also implemented in an attempt to perform adaptive stepsize control. Their accuracy, stability and computational cost are compared through the simulation of classical benchmark problems for the two-dimensional Vlasov–Poisson system.
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References
Banks JW, Hittinger JAF (2010) A new class of nonlinear finite-volume methods for Vlasov simulation. IEEE Trans Plasma Sci 38(9):2198–2207. https://doi.org/10.1109/TPS.2010.2056937
Banks JW, Odu AG, Berger R, Chapman T, Arrighi W, Brunner S (2019) High-order accurate conservative finite difference methods for Vlasov equations in 2D+2V. SIAM J Sci Comput 41(5):B953–B982. https://doi.org/10.1137/19M1238551
Bashforth F, Adams J (1883) An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of fluid. University Press
Bellan PM (2006) Fundamentals of plasma physics. Cambridge University Press, Cambridge
Birdsall CK, Langdon AB (2005) Plasma physics via computer simulation. Taylor & Francis, New York
Boyd TJM, Sanderson JJ (2003) The physics of plasmas. Cambridge University Press, Cambridge
Butcher JC (2008) Numerical methods for ordinary differential equations, 2nd edn. Wiley, Hoboken
Cai X, Guo W, Qiu JM (2018) A high order semi-Lagrangian discontinuous Galerkin method for Vlasov–Poisson simulations without operator splitting. J Comput Phys 354:529–551. https://doi.org/10.1016/j.jcp.2017.10.048
Chen FF (2016) Introduction to plasma physics and controlled fusion. Springer International Publishing, Cham
Cheng Y, Christlieb AJ, Zhong X (2014) Energy-conserving discontinuous Galerkin methods for the Vlasov–Ampère system. J Comput Phys 256:630–655. https://doi.org/10.1016/j.jcp.2013.09.013
Colella P, Dorr M, Hittinger J, Martin D (2011) High-order, finite-volume methods in mapped coordinates. J Comput Phys 230(8):2952–2976. https://doi.org/10.1016/j.jcp.2010.12.044
Conde L (2018) An introduction to plasma physics and its space applications. Morgan & Claypool Publishers, San Rafael
Cottet GH (2018) Semi-Lagrangian particle methods for high-dimensional Vlasov–Poisson systems. J Comput Phys 365:362–375. https://doi.org/10.1016/j.jcp.2018.03.042
Crouseilles N, Mehrenberger M, Sonnendrücker E (2010) Conservative semi-Lagrangian schemes for Vlasov equations. J Comput Phys 229(6):1927–1953. https://doi.org/10.1016/j.jcp.2009.11.007
Crouseilles N, Glanc P, Mehrenberger M, Steiner C (2012) Finite volume schemes for Vlasov. ESAIM Proc 38:275–297. https://doi.org/10.1051/proc/201238015
Dehghan M, Abbaszadeh M (2017) A local meshless method for solving multi-dimensional Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems arising in plasma physics. Eng Comput 33(4):961–981. https://doi.org/10.1007/s00366-017-0509-y
Dehghan M, Mohebbi A (2008) The combination of collocation, finite difference, and multigrid methods for solution of the two-dimensional wave equation: solution of the two-dimensional wave equation. Numer Methods Partial Differ Equ 24(3):897–910. https://doi.org/10.1002/num.20295
Deriaz E, Peirani S (2018) Six-dimensional adaptive simulation of the Vlasov equations using a hierarchical basis. Multiscale Model Simul 16(2):583–614. https://doi.org/10.1137/16M1108649
Fijalkow E (1999) A numerical solution to the Vlasov equation. Comput Phys Commun 116(2–3):319–328. https://doi.org/10.1016/S0010-4655(98)00146-5
Filbet F, Sonnendrücker E, Bertrand P (2001) Conservative numerical schemes for the Vlasov equation. J Comput Phys 172(1):166–187. https://doi.org/10.1006/jcph.2001.6818
Hairer E, Nørsett SP, Wanner G (2009) Solving ordinary differential equations I: nonstiff problems, 2nd rev. ed edn. No. 8 in Springer series in computational mathematics. Springer, Heidelberg
Henrici P (1962) Discrete variable methods in ordinary differential equations. Wiley, New York
Huot F, Ghizzo A, Bertrand P, Sonnendrücker E, Coulaud O (2003) Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov–Maxwell system. J Comput Phys 185(2):512–531. https://doi.org/10.1016/S0021-9991(02)00079-7
Juno J, Hakim A, TenBarge J, Shi E, Dorland W (2018) Discontinuous Galerkin algorithms for fully kinetic plasmas. J Comput Phys 353:110–147. https://doi.org/10.1016/j.jcp.2017.10.009
Lorenzon D, Elaskar SA, Cimino AM (2021) Numerical simulations using Eulerian schemes for the Vlasov–Poisson model. Int J Comput Methods. https://doi.org/10.1142/S0219876221500316
Lorenzon D, Elaskar S, Sánchez-Arriaga G (2016) Simulación Numérica de la Recolección de Corriente en una Sonda de Langmuir Cilíndrica. Mec Comput XXXIV(53):3521 – 3535
Milne WE (1926) Numerical integration of ordinary differential equations. Am Math Mon 33(9):455–460. https://doi.org/10.1080/00029890.1926.11986619
Press WH (ed) (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge
Sánchez-Arriaga G (2013) A direct Vlasov code to study the non-stationary current collection by a cylindrical Langmuir probe. Phys Plasmas 20(1):013504. https://doi.org/10.1063/1.4774398
Sanmartin JR, Martinez-Sanchez M, Ahedo E (1993) Bare wire anodes for electrodynamic tethers. J Propul Power 9(3):353–360. https://doi.org/10.2514/3.23629
Sanmartin JR, Charro M, Pelaez J, Tinao I, Elaskar S, Hilgers A, Martinez-Sanchez M (2006) Floating bare tether as upper atmosphere probe. J Geophys Res. https://doi.org/10.1029/2006JA011624
Vogman G, Colella P, Shumlak U (2014) Dory–Guest–Harris instability as a benchmark for continuum kinetic Vlasov–Poisson simulations of magnetized plasmas. J Comput Phys 277:101–120. https://doi.org/10.1016/j.jcp.2014.08.014
Vogman G, Shumlak U, Colella P (2018) Conservative fourth-order finite-volume Vlasov–Poisson solver for axisymmetric plasmas in cylindrical (r, v, v) phase space coordinates. J Comput Phys 373:877–899. https://doi.org/10.1016/j.jcp.2018.07.029
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This research was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Ministerio de Ciencia, Tecnología e Innovación, Buenos Aires, Argentina.
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Lorenzon, D., Elaskar, S. Using linear multistep methods for the time stepping in Vlasov–Poisson simulations. Comp. Appl. Math. 40, 289 (2021). https://doi.org/10.1007/s40314-021-01683-4
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DOI: https://doi.org/10.1007/s40314-021-01683-4