Pointwise error analysis of the BDF3 compact finite difference scheme for viscous Burgers' equations

Abstract

This paper formulates a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme based on a developed fourth-order operator for computing the approximate solution of the Burger's equation. The equation is one of the useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics and etc. This proposed approach approximates the solution of Burger's equation with the help of two main steps. In the first step, the temporal discretization is accomplished by virtue of the BDF3 approach. In the second step, a developed fourth-order operator and the classic compact difference formula combine with the method of order reduction are applied for spatial discretization, thereby constructing a fully-discrete scheme. The theoretical analysis is proved in detail by means of the discrete energy method. The proposed scheme is convergent with third order for time and fourth order for space. Numerical results are carried out to verify the validity and accuracy of the proposed method.