A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients

TL;DR

This paper introduces a fast, unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations with variable coefficients, achieving high accuracy and efficiency for large-scale problems.

Contribution

It develops a novel numerical scheme combining $L1$-type and WSGD formulas with fast sum-of-exponential approximation for efficient large-scale simulations.

Findings

Achieves $(2 - \gamma)$-order temporal convergence.

Achieves 2-order spatial convergence.

Demonstrates efficiency and effectiveness through numerical experiments.

Abstract

In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Gr\"{u}nwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the $(2 - \gamma)$-order and 2-order of temporal and spatial convergence with $\gamma\in(0,1)$ the order of Caputo fractional derivative, respectively. The fast sum-of-exponential approximation of the generalized Caputo fractional derivative and Toeplitz-like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large-scale simulation of GTSFDEs.