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August  2018, 11(4): 697-714. doi: 10.3934/krm.2018028

A Rosenau-type approach to the approximation of the linear Fokker-Planck equation

Giuseppe Toscani

Department of Mathematics, University of Pavia and IMATI-CNR, Pavia, Italy

Received  April 2017 Revised  May 2017 Published  April 2018

Fund Project: This work has been written within the activities of the National Group of Mathematical Physics (GNFM) of INdAM (National Institute of High Mathematics), and partially supported by the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations".

The numerical approximation of the solution of the Fokker-Planck equation is a challenging problem that has been extensively investigated starting from the pioneering paper of Chang and Cooper in 1970 [8]. We revisit this problem at the light of the approximation of the solution to the heat equation proposed by Rosenau [25]. Further, by means of the same idea, we address the problem of a consistent approximation to higher-order linear diffusion equations.

Keywords: Fokker-Planck equation, discrete schemes, Wild sums, Fourier-based metrics, higher-order diffusions.
Mathematics Subject Classification: 65M06, 65M12, 82C40, 82D10.
Citation: Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
References:
[1]

G. Barbatis and F. Gazzola, Higher order linear parabolic equations, Contemporary Mathematics, 594 (2013), 77-97. doi: 10.1090/conm/594/11775. Google Scholar

[2]

C. Buet and S. Dellacherie, On the Chang and Cooper scheme applied to a linear Fokker-Planck equation, Commun. Math. Sci., 8 (2010), 1079-1090. doi: 10.4310/CMS.2010.v8.n4.a15. Google Scholar

[3]

C. Buet, S. Dellacherie and R. Sentis, Numerical solution of an ionic Fokker-Planck equation with electronic temperature, SIAM J. Numer. Anal., 39 (2001), 1219-1253. doi: 10.1137/S0036142999359669. Google Scholar

[4]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286. doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.0.CO;2-O. Google Scholar

[5]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. Google Scholar

[6]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer Series in Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. Google Scholar

[7]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys., 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1. Google Scholar

[8]

J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck Equation, Journal of Computational Physics, 6 (1970), 1-16. doi: 10.1016/0021-9991(70)90001-X. Google Scholar

[9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. Google Scholar
[10]

S. Dellacherie, Sur un schéma numérique semi-discret appliqué un opérateur de Fokker-Planck isotrope, C.R. Acad. Sci. Paris, série I, 328 (1999), 1219-1224. doi: 10.1016/S0764-4442(99)80443-1. Google Scholar

[11]

S. Dellacherie, Numerical resolution of an ion-electron collision operator in axisymmetrical geometry, Transp. Theory and Stat. Phys., 31 (2002), 397-429. doi: 10.1081/TT-120015507. Google Scholar

[12]

E. M. Epperlein, Implicit and conservative difference scheme for the Fokker-Planck equation, J. Comput. Phys., 112 (1994), 291-297. doi: 10.1006/jcph.1994.1101. Google Scholar

[13]

H. L. Frisch, E. Helfand and J. L. Lebowitz, Nonequilibrium distribution functions in a fluid, Phys. of Fluids, 3 (1960), 325-338. doi: 10.1063/1.1706037. Google Scholar

[14]

G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, On Rosenau-type approximations to fractional diffusion equations, Commun. Math. Sci., 13 (2015), 1163-1191. doi: 10.4310/CMS.2015.v13.n5.a5. Google Scholar

[15]

E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934. doi: 10.1007/BF02179298. Google Scholar

[16]

E. W. Larsen, C. D. Levermore, G. C. Pomraning and J. G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators, Journal of Computational Physics, 61 (1985), 359-390. doi: 10.1016/0021-9991(85)90070-1. Google Scholar

[17]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R. Google Scholar

[18]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33 (2001), 930-945. doi: 10.1137/S0036141001386908. Google Scholar

[19]

M. Mohammadi and A. Borzì, Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations, J. Numer. Math., 23 (2015), 271-288. doi: 10.1515/jnma-2015-0018. Google Scholar

[20]

V. A. Mousseau and D. A. Knoll, Fully implicit kinetic solution of collisional plasmas, J. Comput. Phys., 136 (1997), 308-323. doi: 10.1006/jcph.1997.5736. Google Scholar

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations & Monte Carlo methods, Oxford University Press, Oxford, 2013. Google Scholar
[22]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600, arXiv: 1702.00088v1. doi: 10.1007/s10915-017-0510-z. Google Scholar

[23]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau type approximations to the heat equation, SIAM J. Appl. Math., 73 (2013), 1416-1438. doi: 10.1137/120876290. Google Scholar

[24] H. Risken, The Fokker-Planck equation: Methods of Solution and Applications, 2nd Ed., Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. Google Scholar
[25]

P. Rosenau, Tempered diffusion: A transport process with propagating fronts and inertial delay, Physical Review A, 46 (1992), 12-15. Google Scholar

[26]

G. Toscani, Sur l'inégalité logarithmique de Sobolev, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 689-694. doi: 10.1016/S0764-4442(97)86991-1. Google Scholar

[27]

G. Toscani, The grazing collisions asymptotics of the non cut-off Kac equation, M2AN Math. Model. Numer. Anal., 32 (1998), 763-772. doi: 10.1051/m2an/1998320607631. Google Scholar

[28]

G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation, Quarterly of Applied Mathematics, 57 (1999), 521-541. doi: 10.1090/qam/1704435. Google Scholar

[29]

E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Camb. Phyl. Soc., 47 (1951), 602-609. doi: 10.1017/S0305004100026992. Google Scholar

show all references

References:
[1]

G. Barbatis and F. Gazzola, Higher order linear parabolic equations, Contemporary Mathematics, 594 (2013), 77-97. doi: 10.1090/conm/594/11775. Google Scholar

[2]

C. Buet and S. Dellacherie, On the Chang and Cooper scheme applied to a linear Fokker-Planck equation, Commun. Math. Sci., 8 (2010), 1079-1090. doi: 10.4310/CMS.2010.v8.n4.a15. Google Scholar

[3]

C. Buet, S. Dellacherie and R. Sentis, Numerical solution of an ionic Fokker-Planck equation with electronic temperature, SIAM J. Numer. Anal., 39 (2001), 1219-1253. doi: 10.1137/S0036142999359669. Google Scholar

[4]

J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations, Math. Methods Appl. Sci., 21 (1998), 1269-1286. doi: 10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.0.CO;2-O. Google Scholar

[5]

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198. Google Scholar

[6]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer Series in Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9. Google Scholar

[7]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys., 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1. Google Scholar

[8]

J. S. Chang and G. Cooper, A practical difference scheme for Fokker-Planck Equation, Journal of Computational Physics, 6 (1970), 1-16. doi: 10.1016/0021-9991(70)90001-X. Google Scholar

[9] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. Google Scholar
[10]

S. Dellacherie, Sur un schéma numérique semi-discret appliqué un opérateur de Fokker-Planck isotrope, C.R. Acad. Sci. Paris, série I, 328 (1999), 1219-1224. doi: 10.1016/S0764-4442(99)80443-1. Google Scholar

[11]

S. Dellacherie, Numerical resolution of an ion-electron collision operator in axisymmetrical geometry, Transp. Theory and Stat. Phys., 31 (2002), 397-429. doi: 10.1081/TT-120015507. Google Scholar

[12]

E. M. Epperlein, Implicit and conservative difference scheme for the Fokker-Planck equation, J. Comput. Phys., 112 (1994), 291-297. doi: 10.1006/jcph.1994.1101. Google Scholar

[13]

H. L. Frisch, E. Helfand and J. L. Lebowitz, Nonequilibrium distribution functions in a fluid, Phys. of Fluids, 3 (1960), 325-338. doi: 10.1063/1.1706037. Google Scholar

[14]

G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, On Rosenau-type approximations to fractional diffusion equations, Commun. Math. Sci., 13 (2015), 1163-1191. doi: 10.4310/CMS.2015.v13.n5.a5. Google Scholar

[15]

E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934. doi: 10.1007/BF02179298. Google Scholar

[16]

E. W. Larsen, C. D. Levermore, G. C. Pomraning and J. G. Sanderson, Discretization methods for one-dimensional Fokker-Planck operators, Journal of Computational Physics, 61 (1985), 359-390. doi: 10.1016/0021-9991(85)90070-1. Google Scholar

[17]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R. Google Scholar

[18]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33 (2001), 930-945. doi: 10.1137/S0036141001386908. Google Scholar

[19]

M. Mohammadi and A. Borzì, Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations, J. Numer. Math., 23 (2015), 271-288. doi: 10.1515/jnma-2015-0018. Google Scholar

[20]

V. A. Mousseau and D. A. Knoll, Fully implicit kinetic solution of collisional plasmas, J. Comput. Phys., 136 (1997), 308-323. doi: 10.1006/jcph.1997.5736. Google Scholar

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations & Monte Carlo methods, Oxford University Press, Oxford, 2013. Google Scholar
[22]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600, arXiv: 1702.00088v1. doi: 10.1007/s10915-017-0510-z. Google Scholar

[23]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau type approximations to the heat equation, SIAM J. Appl. Math., 73 (2013), 1416-1438. doi: 10.1137/120876290. Google Scholar

[24] H. Risken, The Fokker-Planck equation: Methods of Solution and Applications, 2nd Ed., Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. Google Scholar
[25]

P. Rosenau, Tempered diffusion: A transport process with propagating fronts and inertial delay, Physical Review A, 46 (1992), 12-15. Google Scholar

[26]

G. Toscani, Sur l'inégalité logarithmique de Sobolev, C. R. Acad. Sci. Paris Sér. I. Math., 324 (1997), 689-694. doi: 10.1016/S0764-4442(97)86991-1. Google Scholar

[27]

G. Toscani, The grazing collisions asymptotics of the non cut-off Kac equation, M2AN Math. Model. Numer. Anal., 32 (1998), 763-772. doi: 10.1051/m2an/1998320607631. Google Scholar

[28]

G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation, Quarterly of Applied Mathematics, 57 (1999), 521-541. doi: 10.1090/qam/1704435. Google Scholar

[29]

E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Camb. Phyl. Soc., 47 (1951), 602-609. doi: 10.1017/S0305004100026992. Google Scholar

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