A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients

Yuan-Ming Wang and Lei Ren

Applied Mathematics and Computation, 2019, vol. 342, issue C, 71-93

Abstract: A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0, 1). The Caputo time-fractional derivative is discretized by a (3−α) th-order numerical formula (called the L2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal (3−α) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results.

Keywords: Fractional sub-diffusion equation; Variable coefficient; Compact difference method; High-order convergence; Energy method (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:342:y:2019:i:c:p:71-93

DOI: 10.1016/j.amc.2018.09.007

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