High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

Abstract

In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order $$ \theta $$ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with $$O(h^{k+1}+\varDelta t^2)$$ is derived, where $$k\geqslant 0$$ denotes the index of the basis function. Extensive numerical results with $$Q^k(k=0,1,2,3)$$ elements are provided to confirm our theoretical results, which also show that the second-order convergence rate in time is not impacted by the changed parameter $$\theta$$ .