Abstract
A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method.
Highlights
Much physical phenomena are described by nonlinear partial differential equations
We have developed a Galerkin linear finite element method to investigate the propagation of solitons and their interactions governed by the nonlinear MRLW equation
An extrapolation technique has been used to improve the accuracy in time for this numerical method
Summary
Much physical phenomena are described by nonlinear partial differential equations. Most of these equations do not have an analytical solution, or it is extremely difficult and expensive to compute their analytical solutions. Where μ is a positive constant, is a nonlinear evolution equation, which was originally introduced by Peregrine [1] in describing the behavior of an undular bore and studied later by Benjamin et al [2]. Some numerical methods [18,19,20,21,22,23,24] for the GRLW equation have been presented, such as a finite difference method [18], a decomposition method [20], and a sinc-collocation method [23] Another special case of the GRLW equation is called the modified regularized long wave (MRLW) equation in which p = 2 [25].
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