A fast compact finite difference scheme for the fourth-order diffusion-wave equation

Abstract

In this paper, the H${}_2$N${}_2$ method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel $t^{1-\gamma }$. Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When $K_1<\frac{3}{2}$, the convergence order is $O({\tau }^{3-\gamma }+h^4+ϵ)$, where $K_1$ is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

Keywords:

• Weakly singular kernel
• H${}_2$N${}_2$ interpolation
• fast algorithm
• stability and convergence

2020 AMS subject classifications:

• 45K05
• 65M06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China Mathematics Tianyuan Foundation [grant numbers 12226337, 12226340, 12126321 and 12126307], Scientific Research Fund of Hunan Provincial Education Department [grant number 21B0550], Hunan Provincial Natural Science Foundation of China [grant number 2022JJ50083].