×

zbMATH — the first resource for mathematics

Finite difference approximations and dynamics simulations for the Lévy fractional Klein-Kramers equation. (English) Zbl 1312.65137
Summary: The Klein-Kramers equation describes position and velocity distribution of Langevin dynamics, the diffusion equation and Fokker-Planck equation are its special cases for characterizing position distribution and velocity distribution, respectively. Incorporating the mechanisms of Lévy flights into the Klein-Kramers formalism leads to the Lévy fractional Klein-Kramers equation, which can effectively describe Lévy flights in the presence of an external force field in the phase space. For numerically solving the Lévy fractional Klein-Kramers equation, this article presents the explicit and implicit finite difference schemes. The discrete maximum principle is generalized, using this result the detailed stability and convergence analyses of the schemes are given. And the extrapolation and some other possible techniques for improving the convergent rate or making the schemes efficient in more general cases are also discussed. The extensive numerical experiments are performed to confirm the effectiveness of the numerical schemes or simulate the superdiffusion processes.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
FODE; ma2dfc
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ott, Anomalous diffusion in ”living polymers”: a genuine Lévy flight?, Phys Rev Lett 65 pp 2201– (1990) · doi:10.1103/PhysRevLett.65.2201
[2] Metzler, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys Rep 339 pp 1– (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[3] Metzler, From a generalized Chapman-Kolmogorov equation to the fractional Klein-Kramers equation, J Phys Chem B 104 pp 3851– (2000) · doi:10.1021/jp9934329
[4] Metzler, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J Phys A: Math Gen 37 pp 161– (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01
[5] Zaslavsky, Chaos, fractional kinetic, and anomalous transport, Phys Rep 371 pp 461– (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[6] Viswanathan, Lévy flight search patterns of wandering albatrosses, Nature 381 pp 413– (1996) · doi:10.1038/381413a0
[7] Viswanathan, Lévy flights and superdiffusion in the context of biological encounters and random searches, Phys Life Rev 5 pp 133– (2008) · doi:10.1016/j.plrev.2008.03.002
[8] Kirchner, Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403 pp 524– (2000) · doi:10.1038/35000537
[9] Shlesinger, Strange kinetics, Nature 363 pp 31– (1993) · doi:10.1038/363031a0
[10] del-Castillo-Negrete, Front dynamics in reaction-diffusion systems with Lévy flights: a fractional diffusion approach, flights in random environments, Phys Rev Lett 93 pp 3699– (2003)
[11] Benson, The fractional-order governing equation of Lévy motion, Water Resour Res 36 pp 1413– (2000) · doi:10.1029/2000WR900032
[12] Chaves, A fractional diffusion equation to describe Lévy flights, Phys Lett A 239 pp 13– (1998) · Zbl 1026.82524 · doi:10.1016/S0375-9601(97)00947-X
[13] Peseckis, Statistical dynamics of stable processes, Phys Rev A 36 pp 892– (1987) · doi:10.1103/PhysRevA.36.892
[14] Brockmann, Lévy flights in external force fields: from models to equations, Chem Phys 284 pp 409– (2002) · doi:10.1016/S0301-0104(02)00671-7
[15] S. Samko A. Kilbas O. Marichev Fractional integrals and derivatives: theory and applications Gordon and Breach London (1993)
[16] Yang, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl Math Modell 34 pp 200– (2010) · Zbl 1185.65200 · doi:10.1016/j.apm.2009.04.006
[17] Lutz, Fractional transport equations for Lévy stable processes, Phys Rev Lett 86 pp 2208– (2001) · doi:10.1103/PhysRevLett.86.2208
[18] Cartling, Kinetics of activated processes from nonstationary solutions of Fokker-Planck equation for a bistable potential, J. Chem Phys 87 pp 2638– (1987) · doi:10.1063/1.453102
[19] H. Risken The Fokker-Planck equation: methods of solution and applications Springer San Diego (1989)
[20] I. Podlubny Fractional differential equations Academic Press San Diego (1999)
[21] Deng, Numerical solution of fractional advection-dispersion different equations, J Hydraul Eng 130 pp 422– (2004) · doi:10.1061/(ASCE)0733-9429(2004)130:5(422)
[22] Liu, Numerical solution of the space fractional Fokker-Planck equation, J Comput Appl Math 166 pp 209– (2004) · Zbl 1036.82019 · doi:10.1016/j.cam.2003.09.028
[23] Meerschaert, Finite different approximations for fractional advection-dispersion flow equations, J Comput Appl Math 172 pp 65– (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[24] Meerschaert, Finite difference approximations for two-sided space-fractional partial differential equations, Appl Numer Math 56 pp 80– (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[25] Deng, Numerical algorithm for the time fractional Fokker-Planck equation, J Comput Phys 227 pp 1510– (2007) · Zbl 1388.35095 · doi:10.1016/j.jcp.2007.09.015
[26] Zhuang, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J Numer Anal 47 pp 1760– (2009) · Zbl 1204.26013 · doi:10.1137/080730597
[27] Zhang, A mass balance based numerical method for fractional advection-dispersion: theory and application, Water Resour Res 41 pp W07029– (2005) · doi:10.1029/2004WR003818
[28] Sousa, Finite difference approximations for a fractional advection diffusion problem, J Comput Phys 228 pp 4038– (2009) · Zbl 1169.65126 · doi:10.1016/j.jcp.2009.02.011
[29] Ervin, Variational formulation for the stationary fractional advection dispersion equation, Numer Methods Partial Differential Equations 22 pp 558– (2005) · Zbl 1095.65118 · doi:10.1002/num.20112
[30] Ervin, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J Numer Anal 45 pp 572– (2007) · Zbl 1141.65089 · doi:10.1137/050642757
[31] Deng, Finite element method for the space and time fractional Fokker-Planck equations, SIAM J Numer Anal 47 pp 204– (2008) · Zbl 1416.65344 · doi:10.1137/080714130
[32] Li, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun Comput Phys 5 pp 1016– (2010) · Zbl 1364.35424
[33] Gorenflo, Discrete random walk models for symmetric Lévy-Feller diffusion processes, Phys A 269 pp 79– (1999) · doi:10.1016/S0378-4371(99)00082-5
[34] Magdziarz, Numerical approach to Klein-Kramers equation, Phys Rev E 76 pp 066708– (2007) · doi:10.1103/PhysRevE.76.066708
[35] Meerschaert, Stochastic solution of space-time fractional diffusion equations, Phys Rev E 65 pp 041103– (2002) · Zbl 1244.60080 · doi:10.1103/PhysRevE.65.041103
[36] Podlubny, Matrix approach to discrete fractional calculus II: partial fractional differential equations, J Comput Phys 228 pp 3137– (2009) · Zbl 1160.65308 · doi:10.1016/j.jcp.2009.01.014
[37] Fok, Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math Comp 71 pp 1497– (2001) · Zbl 1007.65068 · doi:10.1090/S0025-5718-01-01365-5
[38] Deng, Finite difference methods and their physical constraints for the fractional Klein-Kramers equation, Numer Methods Partial Differential Equations 27 pp 1561– (2011) · Zbl 1233.65052 · doi:10.1002/num.20596
[39] Lubich, Discretized fractional calculus, SIAM J Appl Math 17 pp 704– (1986) · Zbl 0624.65015 · doi:10.1137/0517050
[40] T. Ikeda Maximum principle in finite element models for convection-diffusion phenomena 1983 · Zbl 0508.65049
[41] A. A. Samarskiy Theory of finite difference schemes Nauka Moscow (1977)
[42] Tadjeran, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J Comput Phys 220 pp 813– (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[43] Varga, Matrix iterative analysis (2000) · Zbl 1216.65042 · doi:10.1007/978-3-642-05156-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.