Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

Applied Mathematics and Computation

Volumn 217, Issue 12, 2011, Pages 5729-5742

  Chen, Chang Ming ^a   Liu, F ^b   Anh, V ^b   Turner, I ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Convergence; Numerical scheme improving temporal accuracy; Stability; The variable order Galilei invariant advection diffusion equation with a nonlinear source term; The variable order Riemann Liouville fractional partial derivative

Indexed keywords

ADVECTION DIFFUSION EQUATION; CONVERGENCE; FIRST ORDER; NONLINEAR SOURCE TERM; NUMERICAL EXAMPLE; NUMERICAL SCHEME; NUMERICAL SCHEME IMPROVING TEMPORAL ACCURACY; NUMERICAL SIMULATION; SECOND ORDERS; SPATIAL ACCURACY; STABILITY AND CONVERGENCE; THE VARIABLE-ORDER GALILEI INVARIANT ADVECTION DIFFUSION EQUATION WITH A NONLINEAR SOURCE TERM; THE VARIABLE-ORDER RIEMANN-LIOUVILLE FRACTIONAL PARTIAL DERIVATIVE;

ADVECTION; DIFFUSION; FLOW FIELDS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 79551635060     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2010.12.049     Document Type: Article

References (33)

1. B. Baeumer, M. Kovcs, and M. Meerschaert Numerical solutions for fractional reaction-diffusion equations Comput. Math. Appl. 55 2008 2212 2226
2. Chang-Ming Chen, F. Liu, I. Turner, and V. Anh Implicit difference approximation of the Galilei invariant fractional advection diffusion equation ANZIAM J. 48 C775-C789 2007 CTAC2006
3. C.-M. Chen, F. Liu, V. Anh, and I. Turner Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation SIAM J. Sci. Comput. 32 4 2010 1740 1760
4. C.F.M. Coimbra Mechanics with variable-order differential operators Ann. Phys. 12 2003 692 703
5. K.P. Evans, and N. Jacob Feller semigroups obtained by variable order subordination Rev. Mat. Comput. 20 2 2007 293 307
6. R. Gorenflo Discrete random walk models for space-time fractional diffusion Chem.Phys. 284 2002 521 541
7. R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi Time fractional diffusion: a discrete random walk approach Nonlinear Dyn. 29 2002 129 143
8. M. Giona, and H.E. Roman Fractional diffusion equation for transport phenomena in random media Physica A 185 1992 87 97
9. B.I. Henry, and S.L. Wearne Fractional reaction-diffusion Physica A 276 2000 448 455
10. N. Jacob, and H. Leopold Pseudo differential operators with variable order of differentiation generating Feller semigroup Integr. Equat. Oper. Th. 17 1993 544 553
11. K. Kikuchi, and A. Negoro On Markov processes generated by pseudodifferential operator of variable order Osaka J. Math. 34 1997 319 335
12. T. Kosztolowicz Subdiffusion in a system with a thick membrane J. Membrane Sci. 320 2008 492 499
13. T.A.M. Langlands, and B.I. Henry The accuracy and stability of an implicit solution method for the fractional diffusion equation J. Comput. Phys. 205 2005 719 736
14. H.G. Leopold Embedding of function spaces of variable order of differentiation Czech. Math. J. 49 1999 633C644
15. R. Lin, F. Liu, V. Anh, and I. Turner Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation Appl. Math. Comput. 212 2009 435 445
16. F. Liu, C. Yang, and K. Burrage Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term J. Comput. Appl. Math. 231 2009 160 176
17. C.F. Lorenzo, T.T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1998.
18. C.F. Lorenzo, and T.T. Hartley Variable-order and distributed order fractional operators Nonlinear Dyn. 29 2002 57 98
19. R. Metzler, J. Klafter, and I.M. Sokolov Anomalous transport in external fields: continuous time random walksand fractional diffusion equations extended Phys. Rev. E 58 1998 1621 1633
20. R. Metzler, E. Barkai, and J. Klafter Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach Phys. Rev. Lett. 82 1999 3563 3567
21. R. Metzler, and J. Klafter The random walk's guide to anomalous diffusion: a fractional dynamics approach Phys. Rep. 339 2000 1 77
22. R. Metzler, and J. Klafter Boundary value problems for fractional diffusion equations Phys. A 278 2000 107 125
23. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
24. L.E.S. Ramirez, and C.F.M. Coimbra variable order constitutive relation for viscoelasticity Ann. Phys. 16 2007 543 552
25. L.E.S. Ramirez, and C.F.M. Coimbra On the selection and meaning of variable order operators for dynamic modeling Int. J. Differ. 2010 2010 16 Article ID 846107
26. M.D. Ruiz-Medina, V.V. Anh, and J.M. Angulo Fractional generalized random fields of variable order Stoch. Anal. Appl. 22 2 2004 775 799
27. S.G. Samko, and B. Ross Integration and differentiation to a variable fractional order Integr. Transfer Spec. F. 1 1993 277 300
28. C.M. Soon, C.F.M. Coimbra, and M.H. Kobayashi variable viscoelasticity operator Ann. Phys. 14 2005 378 389
29. C. Taddjeran, M. Meerschaert, and H. Scheffler A second-order accurate numerical approximation for the fractional diffusion equation J. Comput. Phys. 213 2006 205 213
30. S.B. Yuste Weighted average finite difference methods for fractional diffusion equations J. Comput. Phys. 216 2006 264 274
31. Y. Zhang, M. Meerschaert, and B. Baeumer Particle tracking for time-fractional diffusion Phys. Rev. E 78 2008 036705
32. P. Zhuang, F. Liu, V. Anh, and I. Turner New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation SIAM J. Numer. Anal. 46 2008 1079 1095
33. P. Zhuang, F. Liu, V. Anh, and I. Turner Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term SIAM J. Numer. Anal. 47 2009 1760 1781