A Fully Discrete LDG Method for the Distributed-Order Time-Fractional Reaction–Diffusion Equation

Abstract

In this paper, the numerical approximation of the distributed-order time-fractional reaction–diffusion equation is proposed and analyzed. Based on the finite difference method in time and local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with order $$O\left( h^{k+\frac{1}{2}}+(\Delta t)^2+\Delta \alpha ^4\right) $$ , where $$h,k,\Delta t$$ and $$\Delta \alpha $$ are the space-step size, piecewise polynomial degree, time-step size, step size in distributed-order variable, respectively. Numerical examples are presented to show the effectiveness and the accuracy of the numerical scheme.