A two-grid block-centered finite difference method for the nonlinear regularized long wave equation

摘要

In this paper, a Crank-Nicolson block-centered finite difference method is first developed and analyzed for the nonlinear regularized long wave equation. By using a cutoff technique, second-order convergences both in time and space are proved under a suitable time-space stepsize constraint condition. To further improve the computational efficiency, an efficient two-grid block-centered finite difference method is introduced and analyzed, in which a resulting small-scale nonlinear problem is first solved on a coarse grid space of size H, and then a resulting large-scale linear problem is solved on a fine grid space of size h. Under a rough time-space stepsize constraint condition Δt = o(H~(1/4)), optimal-order error estimates for both the primal variable and its flux are derived on non-uniform spatial grids. Thus, the proposed method is competitive both in accuracy and efficiency compared with the fully nonlinear Crank-Nicolson block-centered finite difference scheme. Numerical experiments are presented to verify the theoretical analysis.