A Characteristic Finite Element Method for Fractional Convection-Diffusion Equations

Abstract

Abstract: A characteristic finite element method (FEM) is proposed for nonlinear fractional convection-diffusion equations. With the characteristic technique and fractional FEM framework, a fully discrete characteristic finite element scheme is constructed. It is utilized to simulate physical systems and studied in contrast with conventional schemes. It is demonstrated that the method captures exact solutions of governing equations well. The method enjoys good stability and high accuracy even if convection dominates diffusion essentially.

Key words: fractional calculus, convection-diffusion equation, characteristic FEM

CLC Number:

O241.8

ZHU Xiaogang, NIE Yufeng, WANG Jungang, YUAN Zhanbin. A Characteristic Finite Element Method for Fractional Convection-Diffusion Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2017, 34(4): 417-424.

References

[1] ADAMS E E, GELHAR L W. Field study of dispersion in a heterogeneous aquifer:2. Spatial moments analysis[J]. Water Resources Research, 1992, 28(12):3293-3307.
[2] NIGMATULLIN R R. The realization of the generalized transfer equation in a medium with fractal geometry[J]. Physica Status Solidi, 1986, 133(1):425-430.
[3] MEERSCHAERT M M, TADJERAN C. Finite difference approximations for fractional advection-dispersion flow equations[J]. Journal of Computational and Applied Mathematics, 2004, 172(1):65-77.
[4] ERVIN V J, ROOP J P. Variational formulation for the stationary fractional advection dispersion equation[J]. Numerical Methods for Partial Differential Equations, 2006, 22(3):558-576.
[5] TIAN W Y, DENG W H, WU Y J. Polynomial spectral collocation method for space fractional advection-diffusion equation[J]. Numerical Methods for Partial Differential Equations, 2014, 30(2):514-535.
[6] HEJAZI H, MORONEY T, LIU F. Stability and convergence of a finite volume method for the space fractional advection-dispersion equation[J]. Journal of Computational and Applied Mathematics, 2014, 255:684-697.
[7] SOUSA E. Finite difference approximations for a fractional advection diffusion problem[J]. Journal of Computational Physics, 2009, 228(11):4038-4054.
[8] SU L J, WANG W Q, WANG H. A characteristic difference method for the transient fractional convection-diffusion equations[J]. Applied Numerical Mathematics, 2011, 61(8):946-960.
[9] SHEN S, LIU F, ANH V, et al. A characteristic difference method for the variable-order fractional advection-diffusion equation[J]. Journal of Applied Mathematics and Computing, 2013, 42(1-2):371-386.
[10] WANG K X, WANG H. A fast characteristic finite difference method for fractional advection-diffusion equations[J]. Advances in Water Resources, 2011, 34(7):810-816.
[11] XU Q W, HESTHAVEN J S. Discontinuous Galerkin method for fractional convection-diffusion equations[J]. SIAM Journal on Numerical Analysis, 2014, 52(1):405-423.
[12] YILDIRIM A, KOÇAK H. Homotopy perturbation method for solving the space-time fractional advection-dispersion equation[J]. Advances in Water Resources, 2009, 32(12):1711-1716.
[13] MOMANI S, ODIBAT Z. Numerical solutions of the space-time fractional advection-dispersion equation[J]. Numerical Methods for Partial Differential Equations, 2008, 24(6):1416-1429.
[14] JIA H G, NIE Y F, LI J L. Fracture analysis in orthotropic thermoelasticity using extended finite element method[J]. Advances in Applied Mathematics and Mechanics, 2015, 7(6):780-795.
[15] ZHAO G Z, YU X J, LI Z Z. Runge-Kutta control volume discontinuous finite element method for multi-medium field simulations[J]. Chinese Journal of Computational Physics, 2014, 31(3):271-284.
[16] XIE X P, XU J C, XUE G R. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models[J]. Journal of Computational Mathematics, 2008, 26(3):437-455.
[17] DOUGLAS, JR J, RUSSELL T F. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures[J]. SIAM Journal on Numerical Analysis, 1982, 19(5):871-885.
[18] PODLUBNY I. Fractional differential equations[M]. Academic Press, San Diego, CA, 1999.