Resonant Generation of Finite-Amplitude Waves by Flow Past Topography on a β-Plane

The forced Korteweg-de Vries (fKdV) equation is the generic equation for resonant flow past an obstacle. However, for flow past topography on a β-plane, the case when the upstream flow is uniform is anomalous in that there is no quadratic nonlinear term in the fKdV equation. Here we show that in this important case an alternative theory is required and obtain a new evolution equation, which has some similarities to the fKdV equation with two significant differences. These are that a small-amplitude topography now produces finite-amplitude waves and the flow response is limited by a wave breakdown characterized by an incipient flow reversal. Various numerical solutions are described.     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

A meshless method using the radial basis functions for numerical solution of the regularized long wave equation

This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs). To avoid solving the nonlinear system, a predictor‐corrector scheme is proposed. Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

变化底部对非线性表面波的影响

考虑表面张力的作用,研究了不可压缩、无粘性流体流过变化壁面时的共振流动,分析了不同的底部壁面变化对非线性表面波的影响．在导出非线性表面波遵循的fKdV方程后,利用拟谱方法进行数值模拟,用Matlab软件绘制瀑布图,由此得出结论:上凸底部上的波可以看成是向前凸台阶和向后凸台阶分别向前后散射发展的结果,二者不发生相互作用;下凹壁面的波形是向前凹台阶和向后凹台阶相互作用的结果;某些组合式底部的波形是上凸和下凹相互作用的结果．     

Investigation of Coriolis effect on oceanic flows and its bifurcation via geophysical Korteweg–de Vries equation

In this work, we have investigated Coriolis effect on oceanic flows in the equatorial region with the help of geophysical Korteweg–de Vries equation (GKdVE). First, Lie symmetries and conservation laws for the GKdVE have been studied. Later, we implement finite element method for numerical simulations. Propagation of nonlinear solitary structures, their interaction and advancement of solitons can be seen in the results so produced. Additionally, Gaussian initial condition and undular bore initial condition are also investigated. Results so obtained have been found in perfect agreement with the available results. Bifurcation analysis of the oceanic traveling wave of the GKdVE is presented depending on traveling wave velocity and Coriolis parameter. It is discerned that velocity of the traveling wave and Coriolis parameter affect significantly on the propagation of the nonlinear waves.     

The Interaction of a Strong Bore with Small Disturbances in Shallow Water

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.     

Resonant Long–Short Wave Interactions in an Unbounded Rotating Stratified Fluid

A theoretical study is made of finite-amplitude modulated internal wavetrains and the attendant nonlinear interaction with the mean flow induced by the modulations, in an unbounded uniformly stratified Boussinesq fluid rotating around the vertical axis. When the rotation is relatively weak, in particular, 'flat' wavetrains, that feature stronger vertical than horizontal modulations, are resonantly coupled with the mean flow in a manner analogous to the resonant long–short wave interaction between gravity and capillary waves on the surface of deep water. A coupled set of evolution equations for the vertical wavenumber, the wave amplitude, and the mean flow is derived under resonant conditions, and is used to examine the propagation of locally confined wavetrains with initially uniform wavenumber and no pre-existing mean flow. The resonant interaction causes radiation of energy away from a flat wavetrain by means of the induced mean flow which forms a trailing wake; this furnishes a possible mechanism for generating low-frequency inertial–gravity waves in the atmosphere, as suggested by field observations. Moreover, owing to refraction by the mean flow, a finite-amplitude wavetrain may experience rapid wavenumber variations in certain locations, consistent with prior numerical simulations. Eventually, in these regions, the wavenumber tends to become multi-valued, suggesting the formation of caustics.     

Wave Breaking and the Generation of Undular Bores in an Integrable Shallow Water System

The generation of an undular bore in the vicinity of a wave-breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic "cubic" breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Euler–Poisson equation. The motion of the undular bore edges is investigated in detail.     

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.     

Operator splitting for numerical solutions of the RLW Equation

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.     

A multigrid method for the Cahn-Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.     

A bound on oscillations in an unsteady undular bore

Using a model equation for the evolution of long waves at the surface of an incompressible fluid, the number of rapid oscillations of an undular bore is estimated. This estimate relies on the analysis of the Burgers–Korteweg–deVries equation in spaces of analytic functions. Special attention is paid to the effect of viscosity.     

Resonant Flow of a Rotating Fluid Past an Obstacle: The General Case

We consider the flow of a rotating fluid past an antisymmetric obstacle placed on the axis of a cylindrical tube, for the case when the upstream flow is nearly resonant, or critical, so that the speed of a free linear long wave is nearly zero in the frame of reference of the obstacle. The perturbed flow is dominated by the resonant mode, whose amplitude satisfies a forced Korteweg—de Vries equation in this general case when the upstream flow contains radial shear andjor radially dependent angular velocity.     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

Instability of the finite‐difference split‐step method applied to the nonlinear Schrödinger equation. I. standing soliton

We consider numerical instability that can be observed in simulations of solitons of the nonlinear Schrödinger equation (NLS) by a split‐step method (SSM) where the linear part of the evolution is solved by a finite‐difference discretization. The von Neumann analysis predicts that this method is unconditionally stable on the background of a constant‐amplitude plane wave. However, simulations show that the method can become unstable on the background of a soliton. We present an analysis explaining this instability. Both this analysis and the features and threshold of the instability are substantially different from those of the Fourier SSM, which computes the linear part of the NLS by a spectral discretization. For example, the modes responsible for the numerical instability are not similar to plane waves, as for the Fourier SSM or, more generally, in the von Neumann analysis. Instead, they are localized at the sides of the soliton. This also makes them different from “physical” (as opposed to numerical) unstable modes of nonlinear waves, which (the modes) are localized around the “core” of a solitary wave. Moreover, the instability threshold for thefinite‐difference split‐step method is considerably relaxed compared with that for the Fourier split‐step. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1002–1023, 2016     

Horizontal water flow in unsaturated porous media using a fractional integral method with an adaptive time step

Predicting the horizontal groundwater flow in unsaturated porous media is a challenge in many areas of science and engineering. The governing equation associated with this phenomenon is a nonlinear partial differential equation known as the Richards equation. However, the numerical results obtained using this equation can differ substantially from the experimental results. In order to overcome this difficulty, a new version of the Richards equation was proposed recently that considers a time derivative of fractional order. In this study, we present a numerical method for solving this fractional Richards equation. Our method comprises an adaptive time marching scheme that uses Picard iterations to solve the corresponding nonlinear equations. A computational code was implemented for the proposed method using the Scilab programming language. We performed numerical simulations of the anomalous diffusion of water in a white siliceous brick and showed that the numerical results were consistent with the available experimental data.     

A Petrov-Galerkin method for equal width equation

The equal width equation is solved numerically by Petrov-Galerkin method using linear hat function and quadratic B-spline function as trial and test functions respectively. Product approximation has been used in this method. A linear stability analysis of the scheme is shown to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. The Maxwellian initial condition pulse and the development of an undular bore are also studied.