A Petrov?Galerkin finite element scheme for the regularized long wave equation

Abstract

A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor-corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumann's stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties. IntroductionThe regularized long wave (RLW) equation was first proposed by Peregrine [23] for modelling the propagation of unidirectional weakly nonlinear and weakly dispersive water waves. Later on, Benjamin et al. [4] proposed the use of the RLW equation as a preferred alternative to the more classical Korteweg-de Vries (KdV) equation [20], to model a larger class of physical phenomena. These authors showed that the RLW equation is better posed than the KdV equation. Abdulloev et al.[1] numerically exposed the inelastic behaviour of solitary waves modelled by the RLW equation, which results from the fact that this model only possesses three conserved quantities, as demonstrated by Olver [22].Numerical solutions of the RLW equation based on the finite difference method were proposed by several authors (see, e.g. [13,14,19,23]). Spectral methods for the same equation were presented by Ben-Yu and Manoranjan [3] and Sloan [25]. Several finite element schemes based on spline Galerkin techniques have been applied to the RLW equation, (see, e.g. [2,5,8,9,15,16,17,18,24]). Luo and Liu [21] proposed a mixed finite element formulation. These finite element formulations were usually approximations with C 1 continuity. A finite volume method was introduced for a generalized KdV-RLW equation by Bradford and Sanders [7]. Finally, Durán and López-Marcos [11,12] showed the importance of conservative numerical methods for the long run simulation and the solitary wave interactions of the RLW model.Our purpose here is to formulate a Petrov-Galerkin finite element method based on a space-time finite element with C 0 continuity. The weight functions are derived from those proposed by Yu and Heinrich [26] for the convective-diffusion equation. It can be extended to bidimensional, multivariate systems [27] and thus amenable to be extended to the Boussinesq equations [6].The r...

Solitary wave solutions of the MRLW equation using radial basis functions

“…. , we need to determine the N + 1 unknown coefficients λ n j from boundary conditions (5) given at x 0 and x N and collocating U n at the remaining N − 1 distinct uniformly distributed interior points…”

A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

“…Nevertheless, in an earlier work [43] the authors showed that a similar scheme for the RLW equation would display the same leading error terms for both the second and the third corrector steps. The RLW model is the Boussinesq model counterpart for kinematic waves, and therefore one might expect that a similar behaviour will be found within the framework of the latter model.…”

Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

“…Space integration of the equation is achieved with the quartic B-splines by substituting the nodal values U and their derivatives U x , U xx in Eqs. (12) and (13). This yields the following coupled matrix system of the first-order ordinary differential equations:…”

“…The change in water level of magnitude U(x, 0) is centered on x = x c . To make comparisons with earlier studies [8,13,18,19,22,24], the parameters are taken as ε = 1.5, µ = 0.16666667, U 0 = 0.1, x c = 0, a = −36, b = 300, mesh size h = 0.24, and time step t = 0.1, d = 2, 5. Figures 20 and 21 show the developments of the undular bores at time t = 250.…”