On a discrete framework of hypocoercivity for kinetic equations
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- by Alain Blaustein and Francis Filbet
- Math. Comp. 93 (2024), 163-202
- DOI: https://doi.org/10.1090/mcom/3862
- Published electronically: July 19, 2023
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Abstract:
We propose and study a fully discrete finite volume scheme for the linear Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted $L^2$ relative entropy. Then, we adapt the arguments developed in Dolbeault, Mouhot, and Schmeiser [Trans. Amer. Math. Soc. 367 (2015), pp. 3807–3828] based on hypocoercivity methods to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.References
- T. P Armstrong, Numerical studies of the nonlinear Vlasov equation, Phys. Fluids 10 (1967), no. 6, 1269–1280.
- Marianne Bessemoulin-Chatard and Francis Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput. 34 (2012), no. 5, B559–B583. MR 3023729, DOI 10.1137/110853807
- Marianne Bessemoulin-Chatard and Francis Filbet, On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system, J. Comput. Phys. 451 (2022), Paper No. 110881, 28. MR 4354369, DOI 10.1016/j.jcp.2021.110881
- M. Bessemoulin-Chatard and F. Filbet, On the convergence of discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system, SINUM J. Numer. Anal. 61 (2023), no. 4.
- Marianne Bessemoulin-Chatard, Maxime Herda, and Thomas Rey, Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations, Math. Comp. 89 (2020), no. 323, 1093–1133. MR 4063313, DOI 10.1090/mcom/3490
- Alain Blaustein and Francis Filbet, An asymptotic-preserving scheme for the Vlasov-Poisson-Fokker-Planck model, Preprint, 2023.
- Martin Burger, José A. Carrillo, and Marie-Therese Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinet. Relat. Models 3 (2010), no. 1, 59–83. MR 2580954, DOI 10.3934/krm.2010.3.59
- Claire Chainais-Hillairet and Francis Filbet, Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model, IMA J. Numer. Anal. 27 (2007), no. 4, 689–716. MR 2371828, DOI 10.1093/imanum/drl045
- Claire Chainais-Hillairet and Maxime Herda, Large-time behaviour of a family of finite volume schemes for boundary-driven convection-diffusion equations, IMA J. Numer. Anal. 40 (2020), no. 4, 2473–2504. MR 4167053, DOI 10.1093/imanum/drz037
- JS Chang and G Cooper, A practical difference scheme for Fokker-Planck equations, J. Comput. Phys. 6 (1970), no. 1, 1–16.
- Anaïs Crestetto, Nicolas Crouseilles, and Mohammed Lemou, A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling, Commun. Math. Sci. 16 (2018), no. 4, 887–911. MR 3878147, DOI 10.4310/CMS.2018.v16.n4.a1
- G. Dimarco, L. Pareschi, and G. Samaey, Asymptotic-preserving Monte Carlo methods for transport equations in the diffusive limit, SIAM J. Sci. Comput. 40 (2018), no. 1, A504–A528. MR 3765913, DOI 10.1137/17M1140741
- Jean Dolbeault, Clément Mouhot, and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc. 367 (2015), no. 6, 3807–3828. MR 3324910, DOI 10.1090/S0002-9947-2015-06012-7
- Guillaume Dujardin, Frédéric Hérau, and Pauline Lafitte, Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker-Planck equations, Numer. Math. 144 (2020), no. 3, 615–697. MR 4071827, DOI 10.1007/s00211-019-01094-y
- Francis Filbet and Tao Xiong, Conservative discontinuous Galerkin/Hermite spectral method for the Vlasov-Poisson system, Commun. Appl. Math. Comput. 4 (2022), no. 1, 34–59. MR 4391992, DOI 10.1007/s42967-020-00089-z
- Francis Filbet and Maxime Herda, A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure, Numer. Math. 137 (2017), no. 3, 535–577. MR 3712285, DOI 10.1007/s00211-017-0885-7
- Francis Filbet and Shi Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), no. 20, 7625–7648. MR 2674294, DOI 10.1016/j.jcp.2010.06.017
- Francis Filbet, Clément Mouhot, and Lorenzo Pareschi, Solving the Boltzmann equation in $N\log _2N$, SIAM J. Sci. Comput. 28 (2006), no. 3, 1029–1053. MR 2240802, DOI 10.1137/050625175
- Francis Filbet and Claudia Negulescu, Fokker-Planck multi-species equations in the adiabatic asymptotics, J. Comput. Phys. 471 (2022), Paper No. 111642, 28. MR 4490445, DOI 10.1016/j.jcp.2022.111642
- Erich L. Foster, Jérôme Lohéac, and Minh-Binh Tran, A structure preserving scheme for the Kolmogorov-Fokker-Planck equation, J. Comput. Phys. 330 (2017), 319–339. MR 3581469, DOI 10.1016/j.jcp.2016.11.009
- Emmanuil H. Georgoulis, Hypocoercivity-compatible finite element methods for the long-time computation of Kolmogorov’s equation, SIAM J. Numer. Anal. 59 (2021), no. 1, 173–194. MR 4199250, DOI 10.1137/19M1296914
- Laurent Gosse and Giuseppe Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal. 43 (2006), no. 6, 2590–2606. MR 2206449, DOI 10.1137/040608672
- Frédéric Hérau, Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models, Lectures on the analysis of nonlinear partial differential equations. Part 5, Morningside Lect. Math., vol. 5, Int. Press, Somerville, MA, 2018, pp. 119–147. MR 3793016
- James Paul Holloway, Spectral velocity discretizations for the Vlasov-Maxwell equations, Transport Theory Statist. Phys. 25 (1996), no. 1, 1–32. MR 1380029, DOI 10.1080/00411459608204828
- Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. 38 (2000), no. 3, 913–936. MR 1781209, DOI 10.1137/S0036142998347978
- G. Joyce, G. Knorr, and H. K. Meier, Numerical integration methods of the Vlasov equation, J. Comput. Phys. 8 (1971), no. 1, 53–63.
- Mohammed Lemou and Luc Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 31 (2008), no. 1, 334–368. MR 2460781, DOI 10.1137/07069479X
- Jian-Guo Liu and Luc Mieussens, Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit, SIAM J. Numer. Anal. 48 (2010), no. 4, 1474–1491. MR 2684343, DOI 10.1137/090772770
- Lorenzo Pareschi and Thomas Rey, Residual equilibrium schemes for time dependent partial differential equations, Comput. & Fluids 156 (2017), 329–342. MR 3693223, DOI 10.1016/j.compfluid.2017.07.013
- Alessio Porretta and Enrique Zuazua, Numerical hypocoercivity for the Kolmogorov equation, Math. Comp. 86 (2017), no. 303, 97–119. MR 3557795, DOI 10.1090/mcom/3157
- Christian Schmeiser and Alexander Zwirchmayr, Convergence of moment methods for linear kinetic equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 74–88. MR 1654590, DOI 10.1137/S0036142996304516
- J. W. Schumer and J. P. Holloway, Vlasov simulations using velocity-scaled Hermite representations, J. Comput. Phys. 144 (1998), no. 2, 626–661.
- Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141. MR 2562709, DOI 10.1090/S0065-9266-09-00567-5
Bibliographic Information
- Alain Blaustein
- Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062, Toulouse, France
- MR Author ID: 1545831
- ORCID: 0009-0006-3866-2175
- Email: alain.blaustein@math.univ-toulouse.fr
- Francis Filbet
- Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062, Toulouse, France
- MR Author ID: 666920
- Email: francis.filbet@math.univ-toulouse.fr
- Received by editor(s): September 30, 2022
- Received by editor(s) in revised form: April 7, 2023
- Published electronically: July 19, 2023
- Additional Notes: Both authors were partially funded by the ANR Project Muffin (ANR-19-CE46-0004).
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 163-202
- MSC (2020): Primary 82C40; Secondary 65N08, 65N35
- DOI: https://doi.org/10.1090/mcom/3862
- MathSciNet review: 4654619