Communications in Mathematical Sciences
Volume 21 (2023)
Error estimate of the nonuniform $L1$ type formula for the time fractional diffusion-wave equation
Pages: 1707 – 1725
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.
diffusion-wave equation, weak singularity, nonuniform mesh, unconditional convergence
2010 Mathematics Subject Classification
65M06, 65M12, 65M15
The authors would like to acknowledge support by the State Key Program of National Natural Science Foundation of China (No. 11931003, 61833005), the National Natural Science Foundation of China (No. 41974133, 11701081, 11701229, U22B2046), the ZhiShan Youth Scholar Program of SEU, China Postdoctoral Science Foundation (No. 2019M651634), and the High-level Scientific Research foundation for the introduction of talent of Nanjing Institute of Technology (No. YKL201856).
Received 17 September 2022
Received revised 24 November 2022
Accepted 15 December 2022
Published 22 September 2023