Hongqiu Chen

Conservative, discontinuous Galerkin–methods for the generalized Korteweg–de Vries equation

We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg–de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the L2–norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling–wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.

Comparison of quarter-plane and two-point boundary value problems: The KdV-equation

This paper is concerned with the Korteweg-de Vries equation which models unidirectional propagation of small amplitude long waves in dispersive media. The two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of length

Comparison of model equations for small-amplitude long waves

L

Comparison of quarter-plane and two-point boundary value problems: the BBM-equation

of the media of propagation is considered. It is shown that the solution of the two-point boundary value problem converges as

Finite Element Methods for a System of Dispersive Equations

L\rightarrow +\infty

Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations

to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. In addition to its intrinsic interest, our result provides justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem for the KdV equation.

Solitary waves in nonlinear dispersive systems

Abstract Consider a body of water of finite depth under the influence of gravity, bounded below by a flat, impermeable surface. If viscous and surface tension effects are ignored, and assuming that the flow is incompressible and irrotational, the fluid motion is governed by the Euler equations together with suitable boundary conditions on the rigid surfaces and on the air-water interface. In special regimes, the Euler equations admit of simpler, approximate models that describe pretty well the fluid response to a disturbance. In situations where the wavelength is long and the amplitude is small relative to the undisturbed depth, and if the Stokes number is of order one, then various model equations have been derived. Two of the most standard are the KdV-equation (0.1) u t +u x +uu x +u xxx =0 and the RLW-equation (0.2) u t +u x +uu x −u xxt =0. Bona, Pritchard and Scott showed that solutions of these two evolution equations agree to the neglected order of approximation over a long time scale, if the initial disturbance in question is genuinely of small-amplitude and long-wavelength. The same formal argument that allows one to infer (0.2) from (0.1) in small-amplitude, long-wavelength regimes also produces a third equation, namely u t +u x +uu x +u xtt =0. Kruskal, in a wide-ranging discussion of modelling considerations, pointed to this equation as an example that might not accurately describe water waves. Its status has remained unresolved. It is our purpose here to show that the initial-value problem for the latter equation is indeed well posed. Moreover, we show that for small-amplitude, long waves, solutions of this model also agree to the neglected order with solutions of either (0.1) or (0.2) provided the initial data is properly imposed. 1

Well-posedness for regularized nonlinear dispersive wave equations

The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.

Periodic initial-value problem for BBM-equation

The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.