A space-time Petrov-Galerkin method for the two-dimensional regularized long-wave equation

Abstract

In this work, a space-time Petrov-Galerkin (STPG) method is used to numerically analyze the two-dimensional regularized long-wave (RLW) equation. The STPG method is a nonstandard finite element method, that is, both of the spatial and temporal variables of this method are discretized by finite element method. Therefore, it can display the superiority of the finite element method in both spatial and temporal directions. In particular, it is easy to obtain space-time high accuracy and is unconditionally stable, thus this method is very appropriate for solving the nonlinear convection-diffusion equations. We prove the existence and uniqueness of the approximate solution and derive an optimal order error estimate in the maximum norm without requiring any compatibility between the space and time mesh size. In addition, some numerical experiments are presented to validate the accuracy and efficiency of the proposed method. Also, we provide several numerical experiments to confirm the conservation properties of discrete mass, energy and momentum for both the single and double-soliton waves.