L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions

Abstract

This paper aims to numerically study the generalized time-fractional Burgers equation in two spatial dimensions using the L1/LDG method. Here the L1 scheme is used to approximate the time-fractional derivative, i.e., Caputo derivative, while the local discontinuous Galerkin (LDG) method is used to discretize the spatial derivative. If the solution has strong temporal regularity, i.e., its second derivative with respect to time being right continuous, then the L1 scheme on uniform meshes (uniform L1 scheme) is utilized. If the solution has weak temporal regularity, i.e., its first and/or second derivatives with respect to time blowing up at the starting time albeit the function itself being right continuous at the beginning time, then the L1 scheme on non-uniform meshes (non-uniform L1 scheme) is applied. Then both uniform L1/LDG and non-uniform L1/LDG schemes are constructed. They are both numerically stable and the $L^2$ optimal error estimate for the velocity is obtained. Numerical examples support the theoretical analysis.

Funding

National Natural Science Foundation of China(12101266)