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Boussinesq Equations (boussinesq + equation)
Selected AbstractsGlobal existence and blow-up of the solutions for the multidimensional generalized Boussinesq equationMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2007Ying Wang Abstract In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the multidimensional generalized Boussinesq equation are obtained. Furthermore, the blow-up of the solution for the Cauchy problem of the generalized Boussinesq equation is proved. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical simulations of the improved Boussinesq equationNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2010Dursun Irk Abstract In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second-and third-order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L, error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagationNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006Basem S. Attili Abstract We will consider the application of the Adomian decomposition method to approximate the solution of the Boussinesq equation. Both the well-posed and the ill-posed cases will be considered. The results obtained will be compared to the theoretical solution for single soliton wave. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source] Instabilities of Boussinesq models in non-uniform depthINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2009F. Løvholt Abstract The von Neumann method for stability analysis of linear waves in a uniform medium is a widely applied procedure. However, the method does not apply to stability of linear waves in a variable medium. Herein we describe instabilities due to variable depth for different Boussinesq equations, including the standard model by Peregrine and the popular generalization by Nwogu. Eigenmodes are first found for bathymetric features on the grid scale. For certain combinations of Boussinesq formulations and bottom profiles stability limits are found in closed form, otherwise numerical techniques are used for the eigenvalue problems. Naturally, the unstable modes in such settings must be considered to be as much a result of the difference method as of the underlying differential (Boussinesq) equations. Hence, modes are also computed for smooth depth profiles that are well resolved. Generally, the instabilities do not vanish with refined resolution. In some cases convergence is observed and we thus have indications of unstable solutions of the differential equations themselves. The stability properties differ strongly. While the standard Boussinesq equations seem perfectly stable, all the other formulations do display unstable modes. In most cases the instabilities are linked to steep bottom gradients and small grid increments. However, while a certain formulation, based on velocity potentials, is very prone to instability, the Boussinesq equations of Nwogu become unstable only under quite demanding conditions. Still, for the formulation of Nwogu, instabilities are probably inherent in the differential equations and are not a result of the numerical model. Copyright © 2008 John Wiley & Sons, Ltd. [source] Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2005K. S. Erduran Abstract A hybrid scheme composed of finite-volume and finite-difference methods is introduced for the solution of the Boussinesq equations. While the finite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the finite-difference scheme. Fourth-order accuracy in space for the finite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams,Basforth third-order predictor and Adams,Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ,HYWAVE', based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd. [source] Numerical solutions of fully non-linear and highly dispersive Boussinesq equations in two horizontal dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2004David R. Fuhrman Abstract This paper investigates preconditioned iterative techniques for finite difference solutions of a high-order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z -level, removing any practical limitations based on the relative water depth. High-order finite difference approximations are shown to be more efficient than low-order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill-conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix-free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh-independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational effort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight effective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright © 2004 John Wiley & Sons, Ltd. [source] Stability and slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2004Alexander Yu. Abstract The global Galerkin method is applied to the benchmark problem that considers an oscillatory regime of convection of air in a tall two-dimensional rectangular cavity. The three most unstable modes of the linearized system of the Boussinesq equations are studied. The converged values of the critical Rayleigh numbers together with the corresponding oscillation frequencies are calculated for each mode. The oscillatory flow regimes corresponding to each of the three modes are approximated asymptotically. No direct time integration is applied. Good agreement with the previously published results obtained by solution of the time-dependent Boussinesq equations is reported. Copyright © 2004 John Wiley & Sons, Ltd. [source] A coupled lattice BGK model for the Boussinesq equations,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2002Zhaoli Guo Abstract In this paper, a thermal lattice BGK model is developed for the Boussinesq incompressible fluids. The basic idea is to solve the velocity field and the temperature field using two independent lattice BGK equations, respectively, and then combine them into one coupled model for the whole system. The porous plate problem and the two-dimensional natural convection flow in a square cavity with Pr=0.71 and various of Rayleigh numbers are simulated using the model. The numerical results are found to be in good agreement with the analytical solutions or those of previous studies. Copyright © 2002 John Wiley & Sons, Ltd. [source] Tsunami generation and propagation from the Mjølnir asteroid impactMETEORITICS & PLANETARY SCIENCE, Issue 9 2007S. Glimsdal The geological structure resulting from the impact is today known as the Mjølnir crater. The present work attempts to model the generation and the propagation of the tsunami from the Mjølnir impact. A multi-material hydrocode SOVA is used to model the impact and the early stages of tsunami generation, while models based on shallow-water theories are used to study the subsequent wave propagation in the paleo-Barents Sea. We apply several wave models of varying computational complexity. This includes both three-dimensional and radially symmetric weakly dispersive and nonlinear Boussinesq equations, as well as equations based on nonlinear ray theory. These tsunami models require a reconstruction of the bathymetry of the paleo-Barents Sea. The Mjølnir tsunami is characteristic of large bolides impacting in shallow sea; in this case the asteroid was about 1.6 km in diameter and the water depth was around 400 m. Contrary to earthquake- and slide-generated tsunamis, this tsunami featured crucial dispersive and nonlinear effects: a few minutes after the impact, the ocean surface was formed into an undular bore, which developed further into a train of solitary waves. Our simulations indicate wave amplitudes above 200 m, and during shoaling the waves break far from the coastlines in rather deep water. The tsunami induced strong bottom currents, in the range of 30,90 km/h, which presumably caused a strong reworking of bottom sediments with dramatic consequences for the marine environment. [source] | ||||||||||