Angel Duran

Numerical behaviour of stable and unstable solitary waves

In this paper we analyse the behaviour in time of the numerical approximations to solitary wave solutions of the generalized Benjamin-Bona-Mahony equation. This equation possesses an important property: the stability of these solutions depends on their velocity. We identify the error propagation mechanisms in both the stable and unstable case. In particular, we show that in the stable case, numerical methods that preserve some conserved quantities of the problem are more appropriate for the simulation of this kind of solutions.

A Numerical Study of the Stability of Solitary Waves of the Bona-Smith Family of Boussinesq Systems

Abstract In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the behavior of solitary waves of these systems under small and large perturbations with the aim of illuminating their long-time asymptotic stability properties and, in the case of large perturbations, examining, among other, phenomena of possible blow-up of the perturbed solutions in finite time.

Numerical approximation of solitary waves of the Benjamin equation

This paper presents several numerical techniques to generate solitary-wave profiles of the Benjamin equation. The formulation and implementation of the methods focus on some specific points of the problem: on the one hand, the approximation of the nonlocal term is accomplished by Fourier techniques, which determine the spatial discretization used in the experiments. On the other hand, in the numerical continuation procedure suggested by the derivation of the model and already discussed in the literature, several algorithms for solving the nonlinear systems are described and implemented: the Petviashvili method, the Preconditioned Conjugate Gradient Newton method and two Squared-Operator methods. A comparative study of these algorithms is made in the case of the Benjamin equation; Newtons method combined with Preconditioned Conjugate Gradient techniques, emerges as the most efficient. The resulting numerical profiles are shown to have a high accuracy as travelling-wave solutions when they are used as initial conditions in a time-stepping procedure for the Benjamin equation. The paper also explores the generation of multi-pulse solitary waves.

Error propagation in the numerical integration of solitary waves. the regularized long wave equation

Abstract We study the error propagation of time integrators of solitary wave solutions for the regularized long wave equation, u t +u x + 1 2 (u 2 ) x −u xxt =0 , by using a geometric interpretation of these waves as relative equilibria. We show that the error growth is linear for schemes that preserve invariant quantities of the problem and quadratic for ‘nonconservative’ methods. Numerical experiments are presented.

On the Galilean Invariance of Some Nonlinear Dispersive Wave Equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.

A plethora of generalised solitary gravity-capillary water waves

The present study describes, first, an efficient algorithm for computing capillary-gravity solitary waves solutions of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

On the Galilean invariance of some dispersive wave equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.

Efficient computation of capillary-gravity generalized solitary waves

This paper is devoted to the computation of capillary-gravity solitary waves of the irrotational incompressible Euler equations with free surface. The numerical study is a continuation of a previous work in several points: an alternative formulation of the Babenko-type equation for the wave profiles, a detailed description of both the numerical resolution and the analysis of the internal flow structure under a solitary wave. The numerical code used in this study is provided in open source for those interested readers.

Peregrine’s System Revisited

In 1967 D. H. Peregrine proposed a Boussinesq-type model for long waves in shallow waters of varying depth. This prominent paper turned a new leaf in coastal hydrodynamics along with contributions by F. Serre, A. E. Green & P. M. Naghdi and many others since then. Several modern Boussinesq-type systems stem from these pioneering works. In the present work we revise the long wave model traditionally referred to as the Peregrine system. Namely, we propose a modification of the governing equations which is asymptotically similar to the initial model for weakly nonlinear waves, while preserving an additional symmetry of the complete water wave problem. This modification procedure is called the invariantization. We show that the improved system has well conditioned dispersive terms in the swash zone, hence allowing for efficient and stable run-up computations.

On the Numerical Approximation to Generalized Ostrovsky Equations: II

In the present chapter, two numerical procedures to simulate the dynamics of generalized versions of the Ostrovsky equation are presented. First, a numerical method to approximate the corresponding periodic initial-value problem is introduced. The scheme consists of a spatial discretization based on Fourier collocation methods, which is justified by the presence of nonlocal terms. Due to the stiff character of the semidiscretization in space, the time integration is performed with a fourth-order, diagonally implicit Runge-Kutta method, which provides additional theoretical and computational properties. The second point treated in this chapter concerns the solitary-wave solutions of the equations. Their numerical generation is carried out by using a Petviashvili-type method, along with acceleration techniques. The resulting procedure is able to compute both classical and generalized solitary waves in an efficient way. The speed-amplitude relation and the asymptotic behaviour of the waves are studied from the computed profiles.