A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials

doi:10.1007/s42967-019-00027-8

Abstract

Abstract: In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF timestepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

Key words: Spectral methods, Semi-Lagrangian methods, High-order, Hermite functions, Vlasov-Poisson equations, Mass conservation

CLC Number:

Lorella Fatone, Daniele Funaro, Gianmarco Manzini. A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System Based on Fourier, Legendre and Hermite Polynomials[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 333-360.

TrendMD

References

1. Arber, T.D., Vann, R.G.L.:A critical comparison of Eulerian-grid-based Vlasov solvers. J. Comput. Phys. 180, 339-357 (2002)
2. Ayuso, B., Carrillo, J.A., Shu, C.-W.:Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinet. Relat. Models 4(4), 955-989 (2011)
3. Ayuso, B., Carrillo, J.A., Shu, C.-W.:Discontinuous Galerkin methods for the multi-dimensional Vlasov-Poisson problem. Math. Models Methods Appl. Sci. 22(12), 1250042 (2012)
4. Banks, J.W., Hittinger, J.A.F.:A new class of nonlinear finite-volume methods for Vlasov simulation. IEEE Trans. Plasma Sci. 38(9), 2198-2207 (2010)
5. Bernardi, C., Maday, Y.:Spectral methods. In:Ciarlet, P., Lions, J. (eds.) Handbook of Numerical Analysis, pp. 209-486. Elsevier, Amsterdam (1997)
6. Birdsall, C.K., Langdon, A.B.:Plasma Physics via Computer Simulation. Taylor and Francis, New York (2005)
7. Bittencourt, J.A.:Fundamentals of Plasma Physics, 3rd edn. Springer-Verlag, New York (2004)
8. Boyd, J.P.:The rate of convergence of Hermite function series. Math. Comput. 35, 1309-1316 (1980)
9. Boyd, J.P.:Asymptotic coefficients of Hermite function series. J. Comput. Phys. 54, 382-410 (1984)
10. Boyd, J.P.:Chebyshev and Fourier Spectral Methods. Springer, Berlin (1989)
11. Boyd, T.J.M., Sanderson, J.J.:The Physics of Plasmas. Cambridge University Press, Cambridge (2003)
12. Brackbill, J.U.:On energy and momentum conservation in particle-in-cell plasma simulation. J. Comput. Phys. 317, 405-427 (2016)
13. Brigham, E.:The Fast Fourier Transform and Its Applications. Prentice Hall, Upper Saddle River (1988)
14. Camporeale, E., Delzanno, G.L., Bergen, B.K., Moulton, J.D.:On the velocity space discretization for the Vlasov-Poisson system:comparison between Hermite spectral and particle-in-cell methods. Part 2:fully-implicit scheme. Comput. Phys. Commun. 198, 47-58 (2016)
15. Camporeale, E., Delzanno, G.L., Lapenta, G., Daughton, W.:New approach for the study of linear Vlasov stability of inhomogeneous systems. Phys. Plasmas 13(9), 092110 (2006)
16. Canuto, C., Hussaini, M.Y., Quarteroni, A.M., Zang, T.A.J.:Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin Heidelberg (1988)
17. Carrillo, J.A., Vecil, F.:Nonoscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 29(3), 1179-1206 (2007)
18. Chen, G., Chacon, L.:A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell algorithm. Comput. Phys. Commun. 197, 73-87 (2015)
19. Chen, G., Chacon, L., Barnes, D.:An energy- and charge-conserving, implicit, electrostatic particle-incell algorithm. J. Comput. Phys. 230(18), 7018-7036 (2011)
20. Cheng, C.Z., Knorr, G.:The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22(3), 330-351 (1976)
21. Christlieb, A., Guo, W., Morton, M., Qiu, J.-M.:A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations. J. Comput. Phys. 267, 7-27 (2014)
22. Cottet, G.H., Raviart, P.-A.:Particle methods for the one-dimensional Vlasov-Poisson equations. SIAM J. Numer. Anal. 21(1), 52-76 (1984)
23. Crouseilles, N., Respaud, T., Sonnendrücker, E.:A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180(10), 1730-1745 (2009)
24. Delzanno, G.L.:Multi-dimensional, fully-implicit, spectral method for the Vlasov-Maxwell equations with exact conservation laws in discrete form. J. Comput. Phys. 301, 338-356 (2015)
25. Fatone, L., Funaro, D., Manzini, G.:Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov-Poisson system. J. Comput. Phys. 384, 349-375 (2019)
26. Filbet, F.:Convergence of a finite volume scheme for the Vlasov-Poisson system. SIAM J. Numer. Anal. 39(4), 1146-1169 (2001)
27. Filbet, F., Sonnendrücker, E.:Comparison of Eulerian Vlasov solvers. Comput. Phys. Commun. 150(3), 247-266 (2003)
28. Filbet, F., Sonnendrücker, E., Bertrand, P.:Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172(1), 166-187 (2001)
29. Funaro, D.:Polynomial Approximation of Differential Equations. LNP, vol. M8. Springer, New York (1992)
30. Funaro, D., Kavian, O.:Approximation of some diffusion evolution equations in unbounded domains by Hermite functions. Math. Comput. 57, 597-619 (1990)
31. Gajewski, H., Zacharias, K.:On the convergence of the Fourier-Hermite transformation method for the Vlasov equation with an artificial collision term. J. Math. Anal. Appl. 61(3), 752-773 (1977)
32. Glassey, R.:The Cauchy Problem in Kinetic Theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996)
33. Gottlieb, D., Orszag, S. A:Numerical Analysis of Spectral Methods:Theory and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1977)
34. Grad, H.:On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331-407 (1949)
35. Guo, B.-Y.:Spectral Methods and Their Applications. World Scientific, Singapore (1998)
36. Guo, B.-Y.:Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68(227), 1067-1078 (1999)
37. Guo, B.-Y., Shen, J., Xu, C.-L.:Spectral and pseudospectral approximations using Hermite functions:application to the Dirac equation. Adv. Comput. Math. 19(1), 35-55 (2003)
38. Guo, B.-Y., Xu, C.-L.:Hermite pseudospectral method for nonlinear partial differential equations. ESAIM:Math. Modell. Numer. Anal. 34(4), 859-872 (2000)
39. Heath, R.E., Gamba, I.M., Morrison, P.J., Michler, C.:A discontinuous Galerkin method for the Vlasov-Poisson system. J. Comput. Phys. 231(4), 1140-1174 (2012)
40. Holloway, J.P.:Spectral velocity discretizations for the Vlasov-Maxwell equations. Transport Theory Stat. Phys. 25(1), 1-32 (1996)
41. Klimas, A.J.:A numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution. J. Comput. Phys. 50(2), 270-306 (1983)
42. Lapenta, G.:Exactly energy conserving semi-implicit particle in cell formulation. J. Comput. Phys. 334, 349-366 (2017)
43. Lapenta, G., Markidis, S.:Particle acceleration and energy conservation in particle in cell simulations. Phys. Plasmas 18, 072101 (2011)
44. Ma, H., Sun, W., Tang, T.:Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains. SIAM J. Numer. Anal. 43, 58-75 (2005)
45. Manzini, G., Delzanno, G., Vencels, J., Markidis, S.:A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system. J. Comput. Phys. 317, 82-107 (2016)
46. Manzini, G., Funaro, D., Delzanno, G.L.:Convergence of spectral discretizations of the Vlasov- Poisson system. SIAM J. Numer. Anal. 55(5), 2312-2335 (2017)
47. Markidis, S., Lapenta, G.:The energy conserving particle-in-cell method. J. Comput. Phys. 230, 7037-7052 (2011)
48. Parker, J.T., Dellar, P.J.:Fourier-Hermite spectral representation for the Vlasov-Poisson system in the weakly collisional limit. J. Plasma Phys. 81(2), 305810203 (2015)
49. Qiu, J.-M., Russo, G.:A high order multidimensional characteristic tracing strategy for the Vlasov- Poisson system. J. Sci. Comput. 71, 414-434 (2017)
50. Schumer, J.W., Holloway, J.P.:Vlasov simulations using velocity-scaled Hermite representations. J. Comput. Phys. 144(2), 626-661 (1998)
51. Shen, J., Tang, T., Wang, L.-L.:Spectral Methods:Algorithms. Analysis and Applications. Springer Publishing Company, Incorporated, New York (2011)
52. Shen, J., Tang, T., Wang, L.-L.:Spectral Methods. Algorithms, Analysis and Applications. SSCM, vol. 41. Springer, Berlin (2011)
53. Shu, C.-W.:Tvb uniformly high-order schemes for conservation laws. Math. Comput. 49, 105-121 (1987)
54. Shu, C.-W.:Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073- 1084 (1988)
55. Shu, C.-W.:A survey of strong stability preserving high order time discretizations. In:Estep, D., Tavener, S. (eds.) Collected Lectures on the Preservation of Stability Under Discretization, pp. 51-65. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)
56. Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439-471 (1988)
57. Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.:The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149(2), 201-220 (1999)
58. Taitano, E.T., Knoll, D.A., Chacon, L., Chen, G.:Development of a consistent and stable fully implicit moment method for Vlasov-Ampère particle in cell (PIC) system. SIAM J. Sci. Comput. 35(5), S126-S149 (2013)
59. Tang, T.:The Hermite spectral method for Gaussian-type functions. SIAM J. Sci. Comput. 14(3), 594- 606 (1993)
60. Trefethen, L.N:Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000)
61. Vencels, J., Delzanno, G., Manzini, G., Markidis, S., Bo Peng, I., Roytershteyn, V.:SpectralPlasmaSolver:a spectral code for multiscale simulations of collisionless, magnetized plasmas. J. Phys. Confer. Ser. 719(1), 012022 (2016)
62. Vencels, J., Delzanno, G. L., Johnson, A., Bo Peng, I., Laure, E., Markidis, S.:Spectral solver for multi-scale plasma physics simulations with dynamically adaptive number of moments. Procedia Comput. Sci. 51, 1148-1157 (2015)
63. Wollman, S.:On the approximation of the Vlasov-Poisson system by particle methods. SIAM J. Numer. Anal. 37(4), 1369-1398 (2000)
64. Wollman, S., Ozizmir, E.:Numerical approximation of the one-dimensional Vlasov-Poisson system with periodic boundary conditions. SIAM J. Numer. Anal. 33(4), 1377-1409 (1996)
65. Xiang, X.-M., Wang, Z.-Q.:Generalized Hermite approximations and spectral method for partial differential equations in multiple dimensions. J. Sci. Comput. 57, 229-253 (2013)

[1]	Jun Zhu, Chi, Wang Shu. Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study [J]. Communications on Applied Mathematics and Computation, 2020, 2(3): 429-460.
[2]	Junying Cao, Ziqiang Wang, Chuanju Xu. A High-Order Scheme for Fractional Ordinary Diferential Equations with the Caputo-Fabrizio Derivative [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 179-199.
[3]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.