An adaptive moving mesh finite element solution of the Regularized Long Wave equation

TL;DR

This paper introduces an adaptive moving mesh finite element method for solving the RLW equation, improving accuracy and efficiency by dynamically adjusting the mesh based on solution features.

Contribution

It develops a novel moving mesh finite element approach for the RLW equation, handling mixed derivatives through a reformulation and demonstrating second-order convergence.

Findings

Method achieves second-order convergence.

Mesh adapts to evolving solution features.

Successfully models solitary waves and undular bores.

Abstract

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.