Communications in Mathematical Sciences

Volume 21 (2023)

Number 6

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

Pages: 1707 – 1725

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a12

Authors

Hong Sun (Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjin, China; and School of Mathematics, Southeast University, Nanjing, China)

Yanping Chen (School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong, China)

Xuan Zhao (School of Mathematics, Southeast University, Nanjing, China)

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 for 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.

Keywords

diffusion-wave equation, weak singularity, nonuniform mesh, unconditional convergence

2010 Mathematics Subject Classification

65M06, 65M12, 65M15

Received 17 September 2022

Received revised 24 November 2022

Accepted 15 December 2022

Published 22 September 2023