In this paper, we introduce high-order finite volume methods for the multi-term time fractional sub-diffusion equation. The time fractional derivatives are described in Caputo’s sense. By using some operators, we obtain the compact finite volume scheme have high order accuracy. We use a compact operator to deal with spatial direction; then we can get the compact finite volume scheme. It is proved that the finite volume scheme is unconditionally stable and convergent in L∞-norm. The convergence order is O(τ2-α + h4). Finally, two numerical examples are given to confirm the theoretical results. Some tables listed also can explain the stability and convergence of the scheme.
References
[1]
Oustaloup, A. (1991) Commande Robuste d’Ordre Non Entier. Hermès, Paris.
[2]
Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
[3]
Oldham, K.B. and Spanier, J. (1974) Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, Cambridge.
[4]
Samko, S.G, Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia.
[5]
Machado, A.T. (1997) Analysis and Design of Fractional-Order Digital Control Systems. Gordon and Breach Science Publishers, Philadelphia.
[6]
Baleanu, D., Defterli, O. and Agrawal, O.P. (2009) A Central Difference Numerical Scheme for Fractional Optimal Control Problems. Journal of Vibration and Control, 15, 583-597. https://doi.org/10.1177/1077546308088565
[7]
Das, S. (2011) Functional Fractional Calculus. Springer, Berlin. https://doi.org/10.1007/978-3-642-20545-3
[8]
Luchko, Y. and Mainardi, F. (2013) Some Properties of the Fundamental Solution to the Signalling Problem for the Fractional Diffusion-Wave Equation. Open Physics, 11, 666-675. https://doi.org/10.2478/s11534-013-0247-8
[9]
Tomovski, I. and Sandev, T. (2013) Exact Solutions for Fractional Diffusion Equation in a Bounded Domain with Different Boundary Conditions. Nonlinear Dynamics, 71, 671-683. https://doi.org/10.1007/s11071-012-0710-x
[10]
Liu, F., Anh, V. and Turner, I. (2004) Numerical Solution of the Space Fractional Fokker-Planck Equation. Journal of Computational and Applied Mathematics, 166, 209-219. https://doi.org/10.1016/j.cam.2003.09.028
[11]
Tadjeran, C. and Meerschaert, M.M. (2006) Hans-Peter Scheffler, a Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213, 205-213. https://doi.org/10.1016/j.jcp.2005.08.008
[12]
Liu, F.W., Zhuang, P., Anh, V., Turner, I. and Burrage, K. (2007) Stability and Convergence of the Difference Methods for the Space-Time Fractional Advection-Diffusion Equation. Applied Mathematics and Computation, 191, 12-20. https://doi.org/10.1016/j.amc.2006.08.162
[13]
Zhuang, P., Liu, F.W., Anh, V. and Turner, I. (2008) New Numerical Methods for the Time Fractial Sub-Diffusion Equation. SIAM Journal on Numerical Analysis, 46, 1079-1095. https://doi.org/10.1137/060673114
[14]
Liu, F.W., Yang, C. and Burrage, K. (2009) Numerical Method and Analytical Technique of the Modified Anomalous Subdiffusion Equation with a Nonlinear Source Term. Journal of Computational and Applied Mathematics, 231, 160-176. https://doi.org/10.1016/j.cam.2009.02.013
[15]
Yuste, S.B. and Acedo, L. (2005) An Explicit Finite Difference Method and a New Von Neumann-Type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42, 1862-1874. https://doi.org/10.1137/030602666
[16]
Yuste, S.B. (2006) Weighted Average Finite Difference Methods for Fractional Diffusion Equations. Journal of Computational Physics, 216, 264-274. https://doi.org/10.1016/j.jcp.2005.12.006
[17]
Langlands, T.A.M. and Henry, B.I. (2005) The Accuracy and Stability of an Implicit Solution Method for the Fractional Diffusion Equation. Journal of Computational Physics, 205, 719-736. https://doi.org/10.1016/j.jcp.2004.11.025
[18]
Sun, Z.Z. and Wu, X.N. (2006) A Fully Discrete Difference Scheme for a Diffusion-Wave System. Applied Numerical Mathematics, 56, 193-209. https://doi.org/10.1016/j.apnum.2005.03.003
[19]
Gao, G. and Sun, Z.Z. (2011) A Compact Finite Difference Scheme for the Fractional Sub-Diffusion Equations. Journal of Computational Physics, 230, 586-595. https://doi.org/10.1016/j.jcp.2010.10.007
[20]
Zhuang, P., Liu, F.W., Anh, V. and Turner, I. (2009) Stability and Convergence of an Implicit Numerical Method for the Non-Linear Fractional Reaction-Subdiffusion Process. IMA Journal of Applied Mathematics, 74, 645-667. https://doi.org/10.1093/imamat/hxp015
[21]
Zhang, Y., Sun, Z.Z. and Wu, H.W. (2011) Error Estimates of Crank-Nicolson-Type Difference Schemes for the Sub-Diffusion Equation. SIAM Journal on Numerical Analysis, 49, 2302-2322. https://doi.org/10.1137/100812707
[22]
Ervin, V.J. and Roop, J.P. (2006) Variational Formulation for the Stationary Fractional Advection Dispersion Equation. Numerical Methods for Partial Differential Equations, 22, 558-576. https://doi.org/10.1002/num.20112
[23]
Tang, Y. (2021) Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems. American Journal of Computational Mathematics, 11, 53-63. https://doi.org/10.4236/ajcm.2021.111005
[24]
Wang, H., Xu, X., Dou, J., et al. (2022) Local Discontinuous Galerkin Method for the Time-Fractional KdV Equation with the Caputo-Fabrizio Fractional Derivative. Journal of Applied Mathematics and Physics, 10, 1918-1935. https://doi.org/10.4236/jamp.2022.106132
[25]
Rui, H.X. (2008) A Conservative Characteristic Finite Volume Element Method for Solution of the Advection-Diffusion Equation. Computer Methods in Applied Mechanics and Engineering, 197, 3862-3869. https://doi.org/10.1016/j.cma.2008.03.013
[26]
Ji, C.C. and Sun, Z.Z. (2015) A High-Order Compact Finite Difference Scheme for the Fractional Sub-Diffusion Equation. Journal of Scientific Computing, 64, 959-985. https://doi.org/10.1007/s10915-014-9956-4
