In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time fractional derivative in the Caputo sense with the order in (0, 1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.
[1] J.X. Cao, C.P. Li, and Y.Q. Chen, High-order approximation to Caputo derivatives and Caputo- type advection-diffusion equations (II), Fract. Calc. Appl. Anal., 18 (2015), 735–761.
[2] P. Castillo, B. Cockburn, D. Sch¨otzau, and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comput., 71 (2002), 455–478.
[3] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[4] Y. Du, Y. Liu, H. Li, Z. Fang, and S. He, Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation, J. Comput. Phys., 344 (2017), 108– 126.
[5] V. J. Ervin, T. Fu¨hrer, N. Heuer, and M. Karkulik, DPG method with optimal test functions for a fractional advection diffusion equation, J. Sci. Comput., 72(2) (2017), 568–585.
[6] G. H. Gao and Z. Z. Sun, Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions, J. Comput. Phys., 298 (2015), 520–538.
[7] X. M. Gu, T. Z. Huang, C. C. Ji, B. Carpentieri, and A. Alikhanov, Fast iterative method with a second-order implicit difference scheme for time-space fractional convection-diffusion equation, J. Sci. Comput., 72 (2017), 957–985.
[8] C. Huang, M. Stynes, and N. An, Optimal L∞(L2) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem, BIT Numer. Math., 58 (2018), 661–690.
[9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Dif- ferential Equations, Elsevier, Netherlands, 2006.
[10] C. P. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Deriva- tives, SIAM, Philadelphia, 2019.
[11] C. P. Li and Z. Wang, The discontinuous Galerkin finite element method for Caputo- type nonlinear conservation law, Math. Comput. Simulat., 169 (2020), 51–73.
[12] C. P. Li, Q. Yi, and J. Kurths, Fractional convection, J. Comput. Nonlinear Dyn., 13(1) (2018), 011004.
[13] C. P. Li and Q. Yi, Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation, Fract. Calc. Appl. Anal., 21(4) (2018), 1046–1072.
[14] C. P. Li and Q. Yi, Modeling and computing of fractional convection equation, Commun. Appl. Math. Comput., 1 (2019), 565–595.
[15] D. Li, C. Wu, and Z. Zhang, Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80 (2019), 403–419.
[16] H. L. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56(2) (2018), 1112–1133.
[17] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.
[18] K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comput., 78(268) (2009), 1975–1995.
[19] B. Rivi`ere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: The- ory and Implementation, SIAM, Philadelphia, 2008.
[20] M. Stynes, E. O’Riordan, and J. L. Gracia, Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079.
[21] L. L. Wei and Y. N. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model., 38(4) (2014), 1511–1522.
[22] Y. Yan, M. Khan, and N. J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data, SIAM J. Numer. Anal., 56 (2018), 210–227.
[23] Y. Y. Zheng and Z. G. Zhao, The discontinuous Galerkin finite element method for fractional cable equation, Appl. Numer. Math., 115 (2017), 32–41.