High-order compact finite difference schemes for solving the regularized long-wave equation

Abstract

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ-weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments.