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

Jian Chen,Taishan Zeng,Yong Huang

doi:10.1155/2021/5581102

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 L 1 -type formula and space discretized by the multiscale Galerkin method. The optimal convergence orders reach O τ 2 − α + h r in the L 2 norm and O τ 2 − α + h r − 1 in the H 1 norm, respectively, in which τ 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.

Highlights

In this paper, we consider the following time fractional Burgers equation [1,2,3,4,5,6,7]: c 0 Dαt uðx, tÞ + uðx, tÞuxðx, − uxx ðx, = f ðx, tÞ, ðx, ∈Ω, ð1Þ with the initial and boundary conditions, given by uðx, 0Þ = u0ðxÞ, 0 ≤ x ≤ 1, ð2Þ uð0, tÞ = uð1, tÞ = 0, 0 < t ≤ T, where 0 < α < 1,Ω = fðx, tÞ ∣ 0 ≤ x ≤ 1, 0 < t ≤ Tg,u0ðxÞ and f ðx, tÞ are given functions, and the notation
We first present a fully discrete scheme for solving the time fractional Burgers equation with the time approximated by the L1-type formula and the space discretization based on the multiscale Galerkin method
We prove that the multilevel augmentation method (MAM) preserves the same optimal convergence order as the original fully discrete scheme

Summary

Introduction

We consider the following time fractional Burgers equation [1,2,3,4,5,6,7]:. We first present a fully discrete scheme for solving the time fractional Burgers equation with the time approximated by the L1-type formula and the space discretization based on the multiscale Galerkin method. Since the time fractional Burgers equation is a nonlinear differential equation, the fully discrete scheme results in a system of nonlinear algebraic equation at each time step Iteration methods such as the Newton iteration method and the quasi-Newton iteration method are often employed to solve these nonlinear equations. The higher accuracy of the approximate solution is required, the larger dimension of the subspace is needed, and the longer computational time is consumed To overcome this problem, we develop the multilevel augmentation method for solving the fully discrete scheme.

Preliminaries

Fully Discrete Scheme and Convergence

Multilevel Augmentation Method for Solving the Fully Discrete Scheme

Numerical Experiments

Conclusion