Propagation Of Long Surface Waves Research Articles

Numerical and analytical investigation for solutions of fractional Oskolkov–Benjamin–Bona–Mahony–Burgers equation describing propagation of long surface waves

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme.

Galerkin finite element methods for the numerical solution of two classical-Boussinesq type systems over variable bottom topography

We consider two ‘Classical’ Boussinesq type systems modelling two-way propagation of long surface waves in a finite channel with variable bottom topography. Both systems are derived from the 1-d Serre–Green–Naghdi (SGN) system; one of them is valid for stronger bottom variations, and coincides with Peregrine’s system, and the other is valid for smaller bottom variations. We discretize in the spatial variable simple initial–boundary-value problems (ibvp’s) for both systems using standard Galerkin-finite element methods and prove L2 error estimates for the ensuing semidiscrete approximations. We couple the schemes with the 4th order-accurate, explicit, classical Runge–Kutta time-stepping procedure and use the resulting fully discrete methods in numerical simulations of dispersive wave propagation over variable bottoms with several kinds of boundary conditions, including absorbing ones. We describe in detail the changes that solitary waves undergo when evolving under each system over a variety of variable-bottom environments. We assess the efficacy of both systems in approximating these flows by comparing the results of their simulations with each other, with simulations of the SGN-system, and with available experimental data from the literature.

An improved Serre model: Efficient simulation and comparative evaluation

• A new Serre model with improved dispersion characteristics is considered. • A careful development of the new model allows an efficient numerical implementation. • A splitting scheme based on high order finite volume and finite difference is used. • Numerical experiments illustrate the superiority of the proposed model. The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre α, β , presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios. A careful development of the Serre α, β model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre α, β model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.

On numerical simulation of tsunami run‐up

AbstractThe improved adaptive grid method is presented for the numerical modelling of propagation of long surface waves in large water areas and their interaction with coasts. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear effects at propagation of long surface waves in the channels with a variable cross-section

Nonlinear effects at propagation of long surface waves in the channels with a variable cross-section

Nelineynye effekty pri rasprostranenii dlinnykh poverkhnostnykh voln v kanalakh peremennogo poperechnogo secheniya

Nelineynye effekty pri rasprostranenii dlinnykh poverkhnostnykh voln v kanalakh peremennogo poperechnogo secheniya

Solution of 2D Boussinesq systems with freefem++: the flat bottom case

-We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom and which was studied in [7,9,10].We present here a freefem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time.We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature.

Full nonlinear dispersion model of shallow water equations on a rotating sphere

Nonlinear dispersion shallow water equations are derived, which describe propagation of long surface waves on a spherical surface with allowance for rotation of the Earth and mobility of the ocean bottom. Derivation of these equations is based on expanding the solution of hydrodynamic equations on a sphere in small parameters depending on the relative thickness of the water layer and dispersion of surface waves.

Long waves through emergent coastal vegetation

AbstractWe study the effects of emergent coastal forests on the propagation of long surface waves of small amplitude. The forest is idealized by an array of vertical cylinders. Simple models are employed to represent bed friction and to simulate turbulence generated by flow through the tree trunks. A multi-scale (homogenization) analysis similar to that for seepage flows is carried out to deduce the effective equations on the macro-scale. The effective coefficients are calculated by numerically solving the micro-scale problem in a unit cell surrounding one or several cylinders. Analytical and numerical solutions for wave attenuation on the macro-scale for different bathymetries and coastal forest configurations are presented. For a transient incident wave, analytical results are discussed for the damping of a leading tsunami. For comparison series of laboratory data for periodic and transient incident waves are also presented. Good agreement is found even though some of the measured waves are short or nonlinear.

On some Boussinesq systems in two space dimensions: theory and numerical analysis

A three-parameter family of Boussinesq type systems in two space dimensions is consid¬ ered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized by Galerkin-fmite element methods, and error estimates are proved for the resulting continuous time semidiscretizations. Results of numerical experiments are also presented with the aim of studying properties of line solitary waves and expanding wave solutions of these systems. Abstract. A three-parameter family of Boussinesq type systems in two space dimensions is consid¬ ered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized by Galerkin-fmite element methods, and error estimates are proved for the resulting continuous time semidiscretizations. Results of numerical experiments are also presented with the aim of studying properties of line solitary waves and expanding wave solutions of these systems.

The reductive perturbation method and the Korteweg-de Vries hierarchy

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variablesτ1,τ3,τ5, ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitudeζ0 satisfies the KdV equation in the timeτ3, it must satisfy the (2n+1)th order equation of the KdV hierarchy in the timeτn+1, withn=2,3,4, ... As a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms (ζ1,ζ2, ...) of the amplitude can be eliminated whenζ0 is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitudeζ0 satisfies the (2n+ 1)th order equation of the KdV hierarchy in the timeτ2n+1. This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.

The Korteweg–de Vries hierarchy and long water-waves

By using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface-waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion, we show that the Korteweg–de Vries hierarchy equations do play a role in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.

Time-Dependent Surface Elevation of an ice Slope

AbstractBy introducing a coordinate stretching, the governing field equations of the creep flow of a non-Newtonian viscous medium down a uniform slope are solved to determine the differential equation describing the propagation of long surface waves caused by initial disturbances and/or time-dependent accumulation-rate The differential equation for the surface wave depends on the flow law of the non-Newtonian fluid, the boundary condition at the ice-bedrock interface, the bedrock topography and the thickness–wavelength ratio. For moderately long waves and small elevation above the mean thickness the results agree in their essentials with those of the kinematic wave theory and the forward wave equation with a diffusion term is derived, but when improving this by allowing higher elevations the Burger's equation and even more complex equations are obtained. To derive these results Glen’s flow law must be generalized to avoid infinitely fast changes in stress deviators close to zero Strain-rates, The range of applicability of the various equations is discussed.

Time-Dependent Surface Elevation of an ice Slope

AbstractBy introducing a coordinate stretching, the governing field equations of the creep flow of a non-Newtonian viscous medium down a uniform slope are solved to determine the differential equation describing the propagation of long surface waves caused by initial disturbances and/or time-dependent accumulation-rate The differential equation for the surface wave depends on the flow law of the non-Newtonian fluid, the boundary condition at the ice-bedrock interface, the bedrock topography and the thickness–wavelength ratio. For moderately long waves and small elevation above the mean thickness the results agree in their essentials with those of the kinematic wave theory and the forward wave equation with a diffusion term is derived, but when improving this by allowing higher elevations the Burger's equation and even more complex equations are obtained. To derive these results Glen’s flow law must be generalized to avoid infinitely fast changes in stress deviators close to zero Strain-rates, The range of applicability of the various equations is discussed.