Volume 12, Issue 3
Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes

Mingrong Cui

10.4208/nmtma.OA-2018-0046

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 845-866.

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  • Abstract

Finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and L1 approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. Stability of the proposed scheme is given using the discrete energy method. The numerical scheme is O (N− min{2−α,rα}) accurate in time, where α (0 < α < 1) is the order of the fractional time derivative, r is an index of the mesh partition, and it is second order accurate in space. Extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete l2-norm and the l-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.

  • History

Published online: 2019-04

  • AMS Subject Headings

35R11, 65M06, 65M12, 65M15

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