An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation

Abstract

A fourth-order compact finite difference method is proposed in this paper to solve one-dimensional Burgers’ equation. The newly proposed method is based on the Hopf–Cole transformation, which transforms the original nonlinear Burgers’ equation into a linear heat equation, and transforms the Dirichlet boundary condition into the Robin boundary condition. The linear heat equation is then solved by an implicit fourth-order compact finite difference scheme. A compact fourth-order formula is also developed to approximate the Robin boundary conditions, while the initial condition for the heat equation is approximated using Simpson’s rule to maintain overall fourth-order accuracy. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. The numerical results also show that the method is unconditionally stable, as there is no constraint on time step size.