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Preface
A Rosenau-type approach to the approximation of the linear Fokker-Planck equation
Department of Mathematics, University of Pavia and IMATI-CNR, Pavia, Italy |
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 [
References:
[1] |
G. Barbatis and F. Gazzola,
Higher order linear parabolic equations, Contemporary Mathematics, 594 (2013), 77-97.
doi: 10.1090/conm/594/11775. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[7] |
S. Chandrasekhar,
Stochastic problems in physics and astronomy, Rev. Modern Phys., 15 (1943), 1-89.
doi: 10.1103/RevModPhys.15.1. |
[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. |
[9] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[7] |
S. Chandrasekhar,
Stochastic problems in physics and astronomy, Rev. Modern Phys., 15 (1943), 1-89.
doi: 10.1103/RevModPhys.15.1. |
[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. |
[9] |
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
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