Error estimate of the nonuniform $L1$ type formula for the time fractional diffusion-wave equation

TL;DR

This paper develops and analyzes a nonuniform L1-type numerical scheme for the time fractional diffusion-wave equation, proving its unconditional convergence and demonstrating its accuracy through numerical experiments.

Contribution

It introduces a new nonuniform L1-type difference scheme with a rigorous convergence proof and novel analysis of discrete convolution kernels on nonuniform meshes.

Findings

Unconditional convergence of the scheme is proved in L2 norm.

Discrete complementary convolution kernels are shown to be positive definite.

Numerical experiments confirm the scheme's accuracy and efficiency.

Abstract

In this paper, a temporal nonuniform $L1$ type difference scheme is built up for the time fractional diffusion-wave equation with the help of the order reduction technique. The unconditional convergence of the nonuniform difference scheme is proved rigorously in $L^2$ norm. Our main tool is the discrete complementary convolution kernels with respect to the coefficient kernels of the L1 type formula. The positive definiteness of the complementary convolution kernels is shown to be vital to the stability and convergence. To the best of our knowledge, this property is proved at the first time on the nonuniform time meshes. Two numerical experiments are presented to verify the accuracy and the efficiency of the proposed numerical methods.