J. Korean Math. Soc. 2021; 58(3): 553-569
Online first article April 2, 2021 Printed May 1, 2021
https://doi.org/10.4134/JKMS.j190423
Copyright © The Korean Mathematical Society.
Yinan Sun, Tie Zhang
Northeastern University; Northeastern University
In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: $\partial^\beta_tu-\hbox{div}(a\nabla u)=f$, $1<\beta<2$. We first construct a difference formula to approximate $\partial^\beta_tu$ by using an interpolation of derivative type. The truncation error of this formula is of $O(\triangle t^{2+\delta-\beta})$-order if function $u(t)\in C^{2,\delta}[0,T]$ where $0\leq\delta\leq 1$ is the H\"older continuity index. This error order can come up to $O(\triangle t^{3-\beta})$ if $u(t)\in C^3[0,T]$. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the $H^1$-norm and $L_2$-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.
Keywords: Fractional diffusion wave equations, finite difference/finite volume method, unconditional stability, optimal error estimate
MSC numbers: Primary 65M60, 65N30, 65N15
Supported by: This work was supported by the State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds, No. 2013ZCX02
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd