The conservative and fourth-order compact finite difference schemes for regularized long wave equation

Abstract

In this paper, two conservative and fourth-order compact finite difference schemes are proposed and analyzed for solving the regularized long wave (RLW) equation. The first compact finite difference scheme is two-level and nonlinear implicit. The second scheme is three-level and linearized implicit. Conservations of the discrete mass and energy, and unique solvability of the numerical solutions are proved. Convergence and unconditional stability are also derived without any restrictions on the grid ratios by using discrete energy method. The optimal error estimates in norm ‖⋅‖ and ‖⋅‖L∞ are of fourth-order and second-order accuracy for the spatial and temporal step sizes, respectively. Numerical examples are presented to simulate the collision of different solitary waves and support the theoretical analysis.