A Novel Temporal Two-Grid Compact Finite Difference Scheme for the Viscous Burgers’ Equation

Abstract

We present a novel two-grid compact finite difference scheme for the viscous Burgers’ equation in this paper, where the second-order Crank-Nicolson method is used to deal with the time marching, the compact finite difference formula is used to approximate the spatial second-order term, and the nonlinear convection term is discretized using the developed nonlinear fourth-order operator, providing the scheme with both high fourth-order spatial convergence and a low computational cost. The scheme is then established in three steps, with the first step being the construction of a nonlinear coarse-grid compact finite difference scheme that is solved iteratively using a fixed point iterative method, the second step being the application of the Lagrange interpolation formula to obtain a rough solution on the fine grid, and the third step being the development of the linearized fine-grid compact finite difference scheme. We also perform a convergence and stability analysis on the developed scheme, and the results show that the scheme can achieve spatial fourth-order and temporal second-order convergence. Finally, a number of numerical examples are provided to validate the theoretical predictions.