Finite difference/local discontinuous Galerkin method for solving the fractional diffusion-wave equation

TL;DR

This paper introduces a finite difference/local discontinuous Galerkin method for solving fractional diffusion-wave equations, providing stability, convergence analysis, and numerical validation of the approach.

Contribution

The paper develops a new finite difference scheme for fractional derivatives and combines it with a discontinuous Galerkin method, achieving unconditionally stable and convergent solutions.

Findings

Method is unconditionally stable.

Convergence order is O(h^{k+1}+(elta t)^2).

Numerical examples confirm theoretical results.

Abstract

In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and give a semidiscrete scheme in time with the truncation error $O((\Delta t)^2)$, where $\Delta t$ is the time step size. Further we develop a fully discrete scheme for the fractional diffusion-wave equation, and prove that the method is unconditionally stable and convergent with order $O(h^{k+1}+(\Delta t)^{2})$, where $k$ is the degree of piecewise polynomial. Extensive numerical examples are carried out to confirm the theoretical convergence rates.