LDG schemes with second order implicit time discretization for a fractional sub-diffusion equation

Abstract

In this paper, we propose new local discontinuous Galerkin (LDG) schemes for solving a time fractional sub-diffusion equation. The new LDG schemes is constructed rely on the splitting of time fractional derivative and space derivative. The key idea of semi-discrete LDG scheme is to split the time fractional derivative into a weak singular integral and a classic first order derivative. By using second time discretization to approximate the weak singular integral and the classic first order derivative, two fully discrete LDG schemes with second order implicit time discretization for the time fractional diffusion equation are presented. The LDG finite element approximations is used in space variable, and Crank–Nicholson, second order backward differentiation formulas are used for the temporal approximation. The error estimates of the presented semi-discrete and fully-discrete LDG schemes are both analyzed. The analysis results show that our numerical schemes are unconditional stable with the second order accuracy in temporal discretization. Numerical experiments are presented to confirm the theoretical results.