Modeling fully nonlinear shallow-water waves and their interactions with cylindrical structures

Introduction of a time-accurate stabilized finite-element approximation for the numerical investigation of fully nonlinear shallow-water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the triangular elements by the Galerkin method, the fourth-order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The Streamline-Upwind Petrov-Galerkin (SUPG) method with cross-wind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Numerical results obtained for cases representing propagation of solitary waves, collisions of two solitary waves, and wave-structure interactions show fairly good agreement with experimental measurements and other published numerical solutions. The comparisons between results from present fully nonlinear wave model and weekly nonlinear wave model are presented and discussed.     

Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

We study the behavior of nonlinear waves in a two-dimensional medium with density and stress relation that vary periodically in space. Efficient approximate Riemann solvers are developed for the corresponding variable-coefficient first-order hyperbolic system. We present direct numerical simulations of this multiscale problem, focused on the propagation of a single localized perturbation in media with strongly varying impedance. For the conditions studied, we find little evidence of shock formation. Instead, solutions consist primarily of solitary waves. These solitary waves are observed to be stable over long times and to interact in a manner approximately like solitons. The system considered has no dispersive terms; these solitary waves arise due to the material heterogeneity, which leads to strong reflections and effective dispersion.     

Modelling solitary waves of a fifth-order non-linear wave equation

A numerical solution of a fifth-order non-linear dispersive wave equation is set up using collocation of seventh-order B-spline interpolation functions over finite elements. A linear stability analysis shows that this numerical scheme, based on a Crank–Nicolson approximation in time, is unconditionally stable. The method is used to model the behaviour of solitary waves.     

Simulating the movement of bed and suspended loads in the coastal environment

A dynamic computer simulation model is implemented to simulate wave-induced transport of bed and suspended material, and the associated development of nearshore profiles. The model uses the Airy, Stokes', cnoidal and solitary wave theories to compute the height, length, celerity, steepness, breaker indices, and horizontal, veritical, mass drift and orbital velocities of random waves propagating along a flat slope. With shoreward wave propagation both bed and suspended loads are computed at discrete grid points.The FORTRAN '77 simulation program is executed for 3000 iterations, with each iteration corresponding to one twelfth of a tidal cycle. The simulated results demonstrate that sediment movement is controlled by the magnitude of the drift, orbital and horizontal wave velocity components, with wave breaking having a direct influence on the deposition of transported bed load. The shallow water regions of cnoidal and solitary waves are characterized by the presence of high concentrations of suspended material. Over the runlength of the simulation, transport of bed and suspended material cause the initialized profile state to change through a sequence of ill-defined transient states to a final morphological state. Examination of the sequence of profile states indicate weak resemblances of barred and nonbarred profile configurations.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

Generation of internal solitary waves in the Sulu Sea and their refraction by bottom topography studied by ERS SAR imagery and a numerical model

ERS-2 SAR images acquired over the Sulu Sea reveal that there are at least three areas where internal solitary waves are generated: (1) at the sill between Dok Kan Island and Pearl Bank; (2) at the sill between Pearl Bank and Talantam Shoal; and (3) at the sill between Talantam Shoal and Sentry Bank. It is observed that the internal solitary waves generated at different source regions merge into a single solitary wave system. When the solitary waves propagate into shallow water, the distance between the solitary waves in a wave packet decreases. Furthermore, when the water depth decreases in the direction of the soliton's crest line, the crest line is bent towards the shallow water region. These observational facts are explained by a wave refraction model which is based on the Korteweg–de Vries equation which is also valid for large amplitude internal solitary waves provided the pycnocline is sufficiently broad.     

A linearized implicit finite-difference method for solving the equal width wave equation

A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation. The method has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore. The obtained results are compared with other numerical results in the literature. A stability analysis of the scheme is also investigated.     

A Not-a-Knot meshless method using radial basis functions and predictor–corrector scheme to the numerical solution of improved Boussinesq equation

A numerical simulation of the improved Boussinesq (IBq) equation is obtained using collocation and approximating the solution by radial basis functions (RBFs) based on the third-order time discretization. To avoid solving the nonlinear system, a predictor–corrector scheme is proposed and the Not-a-Knot method is used to improve the accuracy in the boundary. The method is tested on two problems taken from the literature: propagation of a solitary wave and interaction of two solitary waves. 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 new scheme presented in this paper.     

Computational models for weakly dispersive nonlinear water waves

Numerical methods for the two- and three-dimensional Boussinesq equations governing weakly nonlinear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by finite element discretization in space. Staggered finite difference schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. Efficient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic finite elements, the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Verification of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a significantly increased computational efficiency when using higher-order schemes and bathymetry-adapted finite element grids.     

二维空间旋转孤立波的相互作用

利用图形分析方法对(2 1)维频散长波方程的旋转孤立波之间的相互作用进行了详细分析,发现了旋转孤立波相互作用产生的一些新的重要非线性现象.这就是,两个旋转孤立波的碰撞是完全非弹性的,它们碰撞之后可以合并成一个旋转孤立波或一个不旋转孤立波,同时可以发生波形转换及性质改变等现象.这些现象的发现,对非线性水波传播与相互作用规律的进一步认识、对非线性水波的控制与利用都具有重要的理论意义.     

A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space-time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.     

Multisymplectic numerical method for the Zakharov system

The Zakharov system is one of the important mathematical models in plasma physics. A multisymplectic pseudospectral discretization for the Zakharov system is presented in this paper. The preservation of discrete normal conservation law is proved theoretically. The propagation and the collision behaviors of the solitary waves are investigated numerically. Numerical results show the present method with advantages such as exponential convergence rate in space. Also it keeps the well preservation of other discrete conserved quantities during long-time numerical calculation.     

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor-corrector finite-difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

基于浅水波理论液体晃动初值问题的数值模拟

针对日益受到关注的液体晃动问题,提出了一种基于浅水波理论的研究方案.该方案采用浅水波理论而非势流理论导出系统控制方程,并通过哈密顿体系表达;利用中心有限差分法和Stormer-Verlet算法进行空间和时间离散;模拟了不同初值条件下的液体晃动情况并对比分析了影响系统非线性响应的主要因素.结果表明,基于浅水波理论能有效解决液体晃动问题;与Euler格式对比,Stormer-Verlet算法精度较高;除共振外对于系统非线性响应的影响容器初始位移比初始速度更显著;非共振情况一定条件下,充液容器运动过程中液体晃动能起到阻尼作用.     

Solving Wick-stochastic water waves using a Galerkin finite element method

A Galerkin finite element approximation of Wick-stochastic water waves is developed and numerically investigated. The problems under study consist of a class of shallow water equations driven by white noise. Random effects may appear in the water free surface or in the bottom topography among others. To perform a rigorous study of stochastic effects in the shallow water equations we employ techniques from Wick calculus. The differentiation respect to time and space along with the product operations are performed in a distribution sense. Using the Wiener-Itô chaos expansion for treating the randomness, the governing equations are transformed into a sequence of deterministic shallow water equations to be solved for each chaos coefficient by standard methods from computational fluid dynamics. In our study, we formulate a finite element method for spatial discretization and a backward Euler scheme for time integration. Once the chaos coefficients are obtained, statistical moments for the stochastic solution are carried out. Numerical results are presented for stochastic water waves in the Strait of Gibraltar.     

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.     

A Discontinuous Spectral Element Model for Boussinesq-Type Equations

We present a discontinuous spectral element model for simulating 1D nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The advective fluxes are calculated using an approximate Riemann solver while the dispersive fluxes are obtained by centred numerical fluxes. Numerical computation of solitary wave propagation is used to prove the exponential convergence.