The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient

Abstract

In this paper, an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient. Since the solution of such an equation in general has a typical weak singularity near the initial time

t=0

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, the time-fractional derivative with order in (0, 1) is discretized by L2-1_σ formula on nonuniform meshes. For the spatial derivative, the local discontinuous Galerkin (LDG) method is employed. A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality. Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.

Keywords

Local discontinuous Galerkin (LDG) method / Nonuniform meshes / L2-1_σ method / Stability analysis / Error estimate / Variable coefficient

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Shanghai University