A Finite Difference/Finite Volume Method for Solving the Fractional Diffusion Wave Equation

J. Korean Math. Soc. Published online April 2, 2021

Yinan Sun and Tie Zhang
Northeastern University, Department of Mathematics, Northeastern University, Shenyang, China

Abstract : 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 Hermite 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.