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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

DOI: 10.4208/nmtma.OA-2018-0046

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

Published online: 2019-04

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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 $L$1 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 $\mathcal{O}$ ($N$−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 $l$2-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.

  • Keywords

Fractional diffusion equation, graded mesh, multi-term, variable coefficients, low regularity, stability and convergence analysis.

  • AMS Subject Headings

35R11, 65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-845, author = {Mingrong Cui}, title = {Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {3}, pages = {845--866}, 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 $L$1 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 $\mathcal{O}$ ($N$−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 $l$2-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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0046}, url = {http://global-sci.org/intro/article_detail/nmtma/13133.html} }
TY - JOUR T1 - Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes AU - Mingrong Cui JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 845 EP - 866 PY - 2019 DA - 2019/04 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2018-0046 UR - https://global-sci.org/intro/article_detail/nmtma/13133.html KW - Fractional diffusion equation, graded mesh, multi-term, variable coefficients, low regularity, stability and convergence analysis. AB -

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 $L$1 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 $\mathcal{O}$ ($N$−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 $l$2-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.

Mingrong Cui. (2019). Finite Difference Schemes for the Variable Coefficients Single and Multi-Term Time-Fractional Diffusion Equations with Non-Smooth Solutions on Graded and Uniform Meshes. Numerical Mathematics: Theory, Methods and Applications. 12 (3). 845-866. doi:10.4208/nmtma.OA-2018-0046
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