L 1 -Multiscale Galerkin’s Scheme with Multilevel Augmentation Algorithm for Solving Time Fractional Burgers’ Equation

Abstract

In this paper, we consider the initial boundary value problem of the time fractional Burgers equation. A fully discrete scheme is proposed for the time fractional nonlinear Burgers equation with time discretized by $L1$ -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach $\mathcal{O}\left({\tau }^{2-\alpha }+h^r\right)$ in the $L^2$ norm and $\mathcal{O}\left({\tau }^{2-\alpha }+h^{r-1}\right)$ in the $H^1$ norm, respectively, in which $\tau$ is the time step size, $h$ is the space step size, and $r$ is the order of piecewise polynomial space. Then, a fast multilevel augmentation method (MAM) is developed for solving the nonlinear algebraic equations resulting from the fully discrete scheme at each time step. We show that the MAM preserves the optimal convergence orders, and the computational cost is greatly reduced. Numerical experiments are presented to verify the theoretical analysis, and comparisons between MAM and Newton’s method show the efficiency of our algorithm.