A Novel Temporal Two-Grid Compact Finite Difference Scheme for the Viscous Burgers’ Equation (Advances in Applied Mathematics and Mechanics, 2024)
We present a novel two-grid compact finite difference scheme for the viscous Burgers’ equation in this paper, where the second-order Crank-Nicolson methodis used to deal with the time marching, the compact finite difference formula is used toapproximate the spatial second-order term, and the nonlinear convection term is discretized using the developed nonlinear fourth-order operator, providing the schemewith both high fourth-order spatial convergence and a low computational cost. Thescheme is then established in three steps, with the first step being the construction of anonlinear coarse-grid compact finite difference scheme that is solved iteratively usinga fixed point iterative method, the second step being the application of the Lagrangeinterpolation 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. Wealso perform a convergence and stability analysis on the developed scheme, and theresults show that the scheme can achieve spatial fourth-order and temporal second-order convergence. Finally, a number of numerical examples are provided to validatethe theoretical predictions.