A finite difference/finite volume method for solving the fractional diffusion wave equation
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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.
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 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.
참고문헌 (26)
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