Open Access Research

Analysis and application of the discontinuous Galerkin method to the RLW equation

Jiří Hozman1* and Jan Lamač12

Author Affiliations

1 Faculty of Science, Humanities and Education, Technical University of Liberec, Studentská 2, Liberec, 461 17, Czech Republic

2 Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Prague, 186 75, Czech Republic

For all author emails, please log on.

Boundary Value Problems 2013, 2013:116  doi:10.1186/1687-2770-2013-116

Published: 7 May 2013

Abstract

In this work, our main purpose is to develop of a sufficiently robust, accurate and efficient numerical scheme for the solution of the regularized long wave (RLW) equation, an important partial differential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward difference formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments confirm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently.

PACS Codes: 02.70.Dh, 02.60 Cb, 02.60.Lj, 03.65.Pm, 02.30.-f.

MSC: 65M60, 65M15, 65M12, 65L06, 35Q53.

Keywords:
discontinuous Galerkin method; regularized long wave equation; backward Euler method; linearization; semi-implicit scheme; a priori error estimates; solitary and periodic wave solutions; experimental order of convergence