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

doi:10.1007/s42967-023-00326-1

摘要/Abstract

摘要： 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, 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.

关键词: Local discontinuous Galerkin (LDG) method, Nonuniform meshes, L2-1_σ method, Stability analysis, Error estimate, 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, 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.

Key words: Local discontinuous Galerkin (LDG) method, Nonuniform meshes, L2-1_σ method, Stability analysis, Error estimate, Variable coefficient

中图分类号:

Qiaoqiao Dai, Dongxia Li. The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1378-1397.

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

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

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