Second-Order Convergence and Unconditional Stability on Crank-Nicolson Scheme for Burgers’ Equation

Abstract

In this paper, we study the numerical solution of one-dimensional Burgers equation with non-homogeneous Dirichlet boundary conditions. This nonlinear problem is converted into the linear heat equation with non-homogeneous Robin boundary conditions by Hopf-Cole transformation. The heat equation is discretized by Crank-Nicolson finite difference scheme, and the fourth-order difference schemes for the Robin conditions are combined with the Crank-Nicolson scheme at two endpoints. The proposed method is proved to be second-order convergent and unconditionally stable. The numerical example supports the theoretical results.