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Journal of Scientific Computing
Published in: Journal of Scientific Computing 2/2018

22-01-2018

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

Authors: Yuan-Ming Wang, Lei Ren

Published in: Journal of Scientific Computing | Issue 2/2018

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Abstract

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order \(\mathcal{O}(\tau ^{r}+h^{4})\), where \(r\ge 2\) is a positive integer and \(\tau \) and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of \(2\le r\le 6\), and the optimal error estimates in the weighted \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.
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Appendix
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Metadata
Title
High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form
Authors
Yuan-Ming Wang
Lei Ren
Publication date
22-01-2018
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 2/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-018-0647-4

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