Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation

TL;DR

This paper develops a high-order compact difference scheme for the mixed-type time-fractional Burgers' equation, rigorously proving its convergence and stability, and validating it through numerical experiments.

Contribution

A novel fourth-order compact difference scheme for the mixed-type time-fractional Burgers' equation is proposed, combining fractional derivative discretization and nonlinear compact operators.

Findings

The scheme is proven to be convergent and stable in the $L^{inity}$-norm.

Numerical experiments confirm the theoretical accuracy and stability.

The method effectively handles the fractional and nonlinear terms of the equation.

Abstract

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers' equation, from which $L_1$-discretization formula is employed to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete compact difference scheme can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability are rigorously proved in the $L^{\infty}$-norm by the energy argument and mathematical induction. Finally, several numerical experiments are provided to verify the theoretical analysis.