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1.
We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM‐type equation. Explicit and implicit–explicit Runge–Kutta‐type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants' conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interactions. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
2.
The frequency or dispersion relation for the discontinuous Galerkin mixed formulation of the 1‐D linearized shallow‐water equations is analysed, using several basic DG mixed schemes. The dispersion properties are compared analytically and graphically with those of the mixed continuous Galerkin formulation for piecewise‐linear bases on co‐located grids. Unlike the Galerkin case, the DG scheme does not exhibit spurious stationary pressure modes. However, spurious propagating modes have been identified in all the present discontinuous Galerkin formulations. Numerical solutions of a test problem to simulate fast gravity modes illustrate the theoretical results and confirm the presence of spurious propagating modes in the DG schemes. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
3.
The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite‐element velocity/surface‐elevation pairs that are used to approximate the linear shallow‐water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0?P1, RT0 and P?P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
4.
This paper presents a parametric finite‐difference scheme concerning the numerical solution of the one‐dimensional Boussinesq‐type set of equations, as they were introduced by Peregrine (J. Fluid Mech. 1967; 27 (4)) in the case of waves relatively long with small amplitudes in water of varying depth. The proposed method, which can be considered as a generalization of the Crank‐Nickolson method, aims to investigate alternative approaches in order to improve the accuracy of analogous methods known from bibliography. The resulting linear finite‐difference scheme, which is analysed for stability using the Fourier method, has been applied successfully to a problem used by Beji and Battjes (Coastal Eng. 1994; 23 : 1–16), giving numerical results which are in good agreement with the corresponding results given by MIKE 21 BW (User Guide. In: MIKE 21, Wave Modelling, User Guide. 2002; 271–392) developed by DHI Software. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
5.
The frequency or dispersion relation for the least‐squares mixed formulation of the shallow‐water equations is analysed. We consider the use of different approximation spaces corresponding to co‐located and staggered meshes, respectively. The study includes the effect of Coriolis, and the dispersion properties are compared analytically and graphically with those of the mixed Galerkin formulation. Numerical solutions of a test problem to simulate slow Rossby modes illustrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
6.
A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
7.
The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??1 approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P1 and P2–P1 pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??1 interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
8.
Dispersion analysis of discrete solutions to the shallow water equations has been extensively used as a tool to define the relationships between frequency and wave number and to determine if an algorithm leads to a dual wave number response and near 2Δx oscillations. In this paper, we explore the application of two‐dimensional dispersion analysis to cluster based and Galerkin finite element‐based discretizations of the primitive shallow water equations and the generalized wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid configurations. It is demonstrated that for various algorithms and grid configurations, contradictions exist between the results of one‐dimensional and two‐dimensional dispersion analysis as a result of subtle changes in the mass matrix. Numerical experiments indicate that the two‐dimensional dispersion analysis correctly predicts the existence and onset of near 2Δx noise in the solution. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
9.
A Fourier analysis has been performed for a class of upwind finite volume schemes, including the study of phase speed, group velocity, damping and dispersion. In the first part, pure gravity waves are investigated. As expected, most upwind schemes lead to a significant damping, but they exhibit a better phase behavior than most centered schemes. In the second part, the Coriolis parameter is considered and the Rossby modes are studied. In this case, all selected upwind schemes lead to a severe damping. The numerical results are also compared with those obtained by using a slope limiter approach. It is concluded that most upwind schemes with or without slope limiters present poor results for an accurate calculation of the Rossby modes. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
10.
A. Duran 《国际流体数值方法杂志》2015,78(2):89-121
In the following lines, we propose a numerical scheme for the shallow‐water system supplemented by topography and friction source terms, in a 2D unstructured context. This work proposes an improved version of the well‐balanced and robust numerical model recently introduced by Duran et al. (J. Comp. Phys., 235 , 565–586, 2013) for the pre‐balanced shallow‐water equations, accounting for varying topography. The present work aims at relaxing the robustness condition and includes a friction term. To this purpose, the scheme is modified using a recent method, entirely based on a modified Riemann solver. This approach preserves the robustness and well‐balanced properties of the original scheme and prevents unstable computations in the presence of low water depths. A series of numerical experiments are devoted to highlighting the performances of the resulting scheme. Simulations involving dry areas, complex geometry and topography are proposed to validate the stability of the numerical model in the neighbourhood of wet/dry transitions. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
11.
A p-type finite element scheme is introduced for the three-dimensional shallow water equations with a harmonic expansion in time. The wave continuity equation formulation is used which decouples the problem into a Helmholtz equation for surface elevation and a momentum equation for horizontal velocity. An exploration of the applicability of p methods to this form of the shallow water problem is presented, with a consideration of the problem of continuity errors. The convergence rates and relative computational efficiency between h- and p- type methods are compared with the use of three test cases representing various degrees of difficulty. A channel test case establishes convergence rates, a continental shelf test case examines a problem with accuracy difficulties at the shelf break, and a field-scale test case examines problems with highly irregular grids. For the irregular grids, adaptive h combined with uniform p refinement was necessary to retain high convergence rates. © 1997 John Wiley & Sons, Ltd. 相似文献
12.
This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth‐order compact, the sixth‐order and eighth‐order SCFDM, and the sixth‐order and eighth‐order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi‐implicit Runge–Kutta method is used. In addition, to control the nonlinear instability, an eighth‐order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high‐order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid‐latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth‐order and eighth‐order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth‐order compact method. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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14.
The appearance of spurious pressure modes in early shallow‐water (SW) models has resulted in two common strategies in the finite element (FE) community: using mixed primitive variable and generalized wave continuity equation (GWCE) formulations of the SW equations. One FE scheme in particular, the P ? P1 pair, combined with the primitive equations may be advantageously compared with the wave equation formulations and both schemes have similar data structures. Our focus here is on comparing these two approaches for a number of measures including stability, accuracy, efficiency, conservation properties, and consistency. The main part of the analysis centres on stability and accuracy results via Fourier‐based dispersion analyses in the context of the linear SW equations. The numerical solutions of test problems are found to be in good agreement with the analytical results. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
15.
In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
16.
A. K. Khe 《Journal of Applied Mechanics and Technical Physics》2003,44(2):180-186
Stationary threedimensional flows of a barotropic liquid in a gravity field are considered. In the shallowwater approximation, the Euler equations are transformed into a system of integrodifferential equations by the EulerLagrange change of coordinates. A system of simplewave equations is obtained, for which the theorem of existence of a solution attached to a given shear flow is proved. As an example, a particular solution analogous to the solution of the problem of a gas flow around a convex angle is given. 相似文献
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18.
This paper presents the results of some studies on the development and application of a finite element method (FEM) with a closed-form solution technique for time discretization. The closed-form solution is based on the eigenvalues/vectors of a coefficient matrix. The method is first applied to the one-dimensional linearized shallow water equations and then extended to the two-dimensional shallow water equations. An attempt is made to improve its efficiency by incorporating time splitting and using the closed-form solution technique only for linear terms. Some case studies of a rectangular channel and harbour are presented to illustrate the satisfactory working of the method. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 953–963, 1997. 相似文献
19.
Finite element approaches generally do not guarantee exact satisfaction of conservation laws especially when Dirichlet‐type boundary conditions are imposed. This article discusses improvement of the global mass conservation property of quasi‐bubble finite element solutions for the shallow water equations, focusing on implementations of the surface‐elevation boundary conditions. We propose two alternative implementations, which are shown by numerical verification to be effective in improving the smoothness of solutions near the boundary and in reducing the mass conservation error. The improvement of the mass conservation property contributes to augmenting the reliability and robustness of long‐term time integrations. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
20.
Richard Comblen Jonathan Lambrechts Jean‐François Remacle Vincent Legat 《国际流体数值方法杂志》2010,63(6):701-724
This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P1 and P?P1 mixed formulations lack convergence for inviscid flows. P?P2 pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献