Analysis of difference algorithms for nonlinear dispersive shallow water models

Abstract

We consider the difference schemes for the one-dimensional versions of nonlinear dispersive shallow water models. We analyse the dissipative and dispersive properties and give the results of numerical calculations.Nonlinear dispersive models hold an intermediate position between the complete equations of fluid motion (the Euler equations of motion or the Cauchy-Poisson problem) and the nonlinear and linear shallow water equations [12]. On the one hand, they are approximate models and may result from an asymptotic expansion of the complete model equations in terms of the nonlineary parameter α=^4 0 /// 0 and the variance β = (// 0 /A 0 ) 2 , where A Q is the mean value of the amplitude, // 0 is the average depth, A O is the wavelength [1,16,17,20]. On the other hand, unlike the nonlinear and linear shallow water equations these equations take into account the relationship between the phase velocities of wave motion and the wave number. This allows us to describe the effects of the wave motion which cannot be described in the framework of the nonlinear and linear shallow water equations.In spite of great interest in the nonlinear dispersive models there are not many results of an investigation of these equations as equations of mathematical physics.Note that there exist exact analytic solutions of a travelling wave type for some nonlinear dispersive models (the Green -Naghdi model [19], the Korteweg-de Vries model [10], the regularized long-wave equation [5], the Kim-Reid-Whittaker model [7]). The continuous dependence of the solution on the initial data is investigated for the Bonna-Smith model in [4].Therefore the method of numerical calculations is one of the most frequently used methods of obtaining the solution.The nonlinear dispersive models can be divided into several groups by the dispersive relation. The first model in [16] and the Peregrine second model in [17] have the same dispersive relation which allows imaginary values of the frequency. Stable difference algorithms are not known for them [8].The stable difference algorithms for nonlinear dispersive models, in which the frequency ω is a real function of the wave number K, are described in the literature. These are nonlinear dispersive models by Korteweg-de Vries [3], Long [11], Green-Naghdi [6], Pelinovsky-Zheleznyak [20], Aleshkov [13], Peregrine [17], Kim-Reid-Whittaker [15], the regularized long-wave equation [5]. The nonlinear dispersive models by Boussinesq, Kim-Reid-Whittaker, Peregrine (first model), Green-Naghdi, Bazdenkov-Morozov-Pogutstse [2],

Methods of construction and the analysis of difference schemes for nonlinear dispersive models of wave hydrodynamics

“…In particular, these ideas were used in [1,4,7,8] (see also the references therein). In order to construct numerical algorithms we use the notation of the original systems of equations, which allows us to construct difference splitting schemes convenient for the solution.…”

“…Then in order to avoid iterations we must combine the first and second steps of the algorithm, which is achieved by eliminating Θ. If we choose an explicit approximation, we may introduce artificial dissipation or introduce the recalculation of the continuity equation into the algorithm in order to achieve stability (by analogy with the algorithms in [7,8,10] (4) Finally, by analogy with the second step from the equation we calculate . .…”

New Algorithm for Numerical Simulation of Surface Waves Within the Framework of the Full Nonlinear Dispersive Model

“…The numerical algorithms based on splitting of NLD-equations on the system of ODE's and the elliptic equation, was first proposed in [15] for the unidirectional scalar equation, and then was extended to systems of WNLD-and FNLD-equations [16,17,18,19,20]. The feature of solving systems of NLD-equations is that the group of terms describing the dispersion contains the time derivative of the velocity u (in one-dimensional case).…”

Analysis of the numerical algorithms for nonlinear dispersive shallow-water models

“…This paper is a continuation of the paper dealing with the analysis of the difference schemes of nonlinear dispersive models for the one-dimensional case [7]. Following [7] we analyse the dispersive relations of the nonlinear dispersive models (NDMs) for the two-dimensional case.…”

“…Following [7] we analyse the dispersive relations of the nonlinear dispersive models (NDMs) for the two-dimensional case. The analysis allows us to recognize at least three classes.…”