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1.
A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials. 相似文献
2.
A finite volume scheme for the global shallow water model on the cubed-sphere mesh is proposed and studied in this paper. The new cell-centered scheme is based on Osher’s Riemann solver together with a high-order spatial reconstruction. On each patch interface of the cubed-sphere only one layer of ghost cells is needed in the scheme and the numerical flux is calculated symmetrically across the interface to ensure the numerical conservation of total mass. The discretization of the topographic term in the equation is properly modified in a well-balanced manner to suppress spurious oscillations when the bottom topography is non-smooth. Numerical results for several test cases including a steady-state nonlinear geostrophic flow and a zonal flow over an isolated mountain are provided to show the flexibility of the scheme. Some parallel implementation details as well as some performance results on a parallel supercomputer with more than one thousand processor cores are also provided. 相似文献
3.
Nouh Izem Fayssal Benkhaldoun Slah Sahmim Mohammed Seaid Mohamed Wakrim 《Journal of Applied Mathematics and Computing》2014,44(1-2):467-489
This paper is devoted to solve the system of partial differential equations governing the flow of two superposed immiscible layers of shallow water flows. The system contains source terms due to bottom topography, wind stresses, and nonconservative products describing momentum exchange between the layers. The presence of these terms in the flow model forms a nonconservative system which is only conditionally hyperbolic. In addition, two-layer shallow water flows are often accompanied with moving discontinuities and shocks. Developing stable numerical methods for this class of problems presents a challenge in the field of computational hydraulics. To overcome these difficulties, a new composite scheme is proposed. The scheme consists of a time-splitting operator where in the first step the homogeneous system of the governing equations is solved using an approximate Riemann solver. In the second step a finite volume method is used to update the solution. To remove the non-physical oscillations in the vicinity of shocks a nonlinear filter is applied. The method is well-balanced, non-oscillatory and it is suitable for both low and high values of the density ratio between the two layers. Several standard test examples for two-layer shallow water flows are used to verify high accuracy and good resolution properties for smooth and discontinuous solutions. 相似文献
4.
Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems
We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method. 相似文献
5.
N. M. Borisova V. V. Ostapenko 《Computational Mathematics and Mathematical Physics》2006,46(7):1254-1276
A numerical algorithm is proposed for simulating the propagation of discontinuous waves over a dry bed governed by the shallow water equations in the first approximation. The algorithm is based on a modified conservation law of total momentum that takes into account the concentrated momentum loss associated with the formation of local eddy structures within the framework of the long-wave approximation. The modified conservation law involves a heuristic parameter that is chosen so as to agree with laboratory experiments. Numerical results are presented for the formation, propagation, and transformation of a discontinuous wave arising in a complete or partial (in the planned case) collapse of a dam over a bed with a horizontal or sloping bottom or a bottom with a local obstacle in the tailwater area. 相似文献
6.
Yi Liu Jianzhong Zhou Lixiang Song Qiang Zou Li Liao Yueran Wang 《Applied Mathematical Modelling》2013
A well-balanced Godunov-type finite volume algorithm is developed for modelling free-surface shallow flows over irregular topography with complex geometry. The algorithm is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. Unstructured triangular grids are used to achieve the adaptability of the grid to the geometry of the problem and to facilitate localised refinement. The numerical fluxes are calculated using HLLC approximate Riemann solver, and the MUSCL-Hancock predictor–corrector scheme is adopted to achieve the second-order accuracy both in space and in time where the solutions are continuous, and to achieve high-resolution results where the solutions are discontinuous. The novelties of the algorithm include preserving well-balanced property without any additional correction terms and the wet/dry front treatments. The good performance of the algorithm is demonstrated by comparing numerical and theoretical results of several benchmark problems, including the preservation of still water over a two-dimensional hump, the idealised dam-break flow over a frictionless flat rectangular channel, the circular dam-break, and the shock wave from oblique wall. Besides, two laboratory dam-break cases are used for model validation. Furthermore, a practical application related to dam-break flood wave propagation over highly irregular topography with complex geometry is presented. The results show that the algorithm can correctly account for free-surface shallow flows with respect to its effectiveness and robustness thus has bright application prospects. 相似文献
7.
Huazhong Tang Tao Tang Kun Xu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2004,55(3):365-382
In this paper, the Kinetic Flux Vector Splitting (KFVS)
scheme is extended to solving the shallow water equations with
source terms. To develop a well-balanced scheme between the source
term and the flow convection, the source term effect is accounted
in the flux evaluation across cell interfaces. This leads to a
modified gas-kinetic scheme with particular application to the
shallow water equations with bottom topography. Numerical
experiments show better resolution of the unsteady solution than
conventional finite difference method and KFVS method with little
additional cost. Moreover, some positivity properties of the
gas-kinetic scheme is established. 相似文献
8.
We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom topography. The designed first-and secondorder schemes are tested on a number of numerical examples, in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states. 相似文献
9.
V. M. Goloviznin V. A. Isakov 《Computational Mathematics and Mathematical Physics》2017,57(7):1140-1157
The CABARET scheme is used for the numerical solution of the one-dimensional shallow water equations over a rough bottom. The scheme involves conservative and flux variables, whose values at a new time level are calculated by applying the characteristic properties of the shallow water equations. The scheme is verified using a series of test and model problems. 相似文献
10.
Andrew Bernstein Alina Chertock Alexander Kurganov 《Bulletin of the Brazilian Mathematical Society》2016,47(1):91-103
Finite-volume central-upwind schemes for shallow water equations were proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), 133–160]. These schemes are capable of maintaining “lake-at-rest” steady states and preserving the positivity of the computed water depth. The well-balanced and positivity preserving features of the central-upwind schemes are achieved, in particular, by using continuous piecewise linear interpolation of the bottom topography function. However, when the bottom function is discontinuous or a model with a moving bottom topography is studied, the continuous piecewise linear approximationmay not be sufficiently accurate and robust. 相似文献
11.
Mai Duc Thanh 《Communications in Nonlinear Science & Numerical Simulation》2013,18(2):417-433
This paper deals with numerical treatments for the shallow water equations with discontinuous topography when the initial data belong to both supersonic region and subsonic region. This kind of data are present in both engineering and rivers, but they are not always well-treated in existing schemes. Our goal is to improve the well-balanced scheme constructed earlier in our work by introducing a computing corrector into the construction of the scheme. First, a further study in the construction of the well-balanced scheme reveals that the errors could make the approximate states near the critical surface that ought to be in one side of the critical surface fall into the other side. This qualitative change, though small, may cause much larger errors following stationary hydraulic jumps formed from these approximate states due to the jump of the bottom. Then, we introduce a corrector in the computing algorithm that selects the equilibrium states in the construction of the well-balanced scheme such that the approximate stationary hydraulic jumps always remain in the right region. Numerical tests show that the well-balanced method using an underlying numerical flux such as Lax–Friedrichs flux, FORCE, GFORCE, or Roe fluxes can approximate very well the exact solution even when the initial data are on both supercritical region and subcritical region. 相似文献
12.
The finite volume scheme of Vijayasundaram and Osher-Solomon type for shallow water equations are proposed. The numerical results with discontinuous initial condition and the comparison with Lax-Friedrichs numerical flux are presented for homogeneous case. The extension of the method for the inhomogeneous case is described. 相似文献
13.
A numerical method based on a second-order accurate Godunov-type scheme is described for solving the shallow water equations on unstructured triangular-quadrilateral meshes. The bottom surface is represented by a piecewise linear approximation with discontinuities, and a new approximate Riemann solver is used to treat the bottom jump. Flows with a dry sloping bottom are computed using a simplified method that admits negative depths and preserves the liquid mass and the equilibrium state. The accuracy and performance of the approach proposed for shallow water flow simulation are illustrated by computing one- and two-dimensional problems. 相似文献
14.
15.
We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non-zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method. 相似文献
16.
An Efficient Method for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations 总被引:3,自引:0,他引:3
ShiJin XinWen 《计算数学(英文版)》2004,22(2):230-249
We propose a simple numerical method for calculating both unsteady and steady state solution of hyperbolic system with geometrical source terms having concentrations. Physical problems under consideration include the shallow water equations with topography,and the quasi one-dimensional nozzle flows. We use the interface value, rather than the cell-averages, for the source terms, which results in a well-balanced scheme that can capture the steady state solution with a remarkable accuracy. This method approximates the source terms via the numerical fluxes produced by an (approximate) Riemann solver for the homogeneous hyperbolic systems with slight additional computation complexity using Newton‘s iterations and numerical integrations. This method solves well the subor super-critical flows, and with a transonic fix, also handles well the transonic flows over the concentration. Numerical examples provide strong evidence on the effectiveness of this new method for both unsteady and steady state calculations. 相似文献
17.
In this paper, we propose a robust finite volume scheme to numerically solve
the shallow water equations on complex rough topography. The major difficulty of this
problem is introduced by the stiff friction force term and the wet/dry interface tracking.
An analytical integration method is presented for the friction force term to remove the
stiffness. In the vicinity of wet/dry interface, the numerical stability can be attained by
introducing an empirical parameter, the water depth tolerance, as extensively adopted
in literatures. We propose a problem independent formulation for this parameter, which
provides a stable scheme and preserves the overall truncation error of $\mathbb{O}$∆$x^3$. The
method is applied to solve problems with complex rough topography, coupled with $h$-adaptive mesh techniques to demonstrate its robustness and efficiency. 相似文献
18.
We consider a two-dimensional shallow water system over movable beds. We begin with a continuous system and prove the existence of the solutions, and then we investigate their smoothness. Then, we employ a Galerkin method to obtain a finite-dimensional problem which is solved using a Brouwer fixed point theorem. Therefore, we show that the limits of the resulting solution sequences satisfy the model equations.After solving the continuous problem, we focus on the corresponding discrete problem. We employ a local discontinuous Galerkin scheme for numerical solution of the discrete system and conduct an error analysis of the numerical scheme. We prove that the method is convergent and that the error is bounded according to a specific norm defined herein. 相似文献
19.
Nina Shokina 《PAMM》2010,10(1):653-654
The numerical modelling of surface water waves generated by a moving underwater landslide on irregular bottom is considered. The non-linear shallow water model is used with taking into account bottom mobility. The equations are obtained for an underwater landslide movement under the action of gravity force, buoyancy force, friction force and water resistance force. The predictor-corrector scheme [5], preserving the monotonicity of the numerical solution profiles in a linear case, is used on adaptive grids. The scheme is validated on the problem with a known analytical solution. The analysis is done for the dependencies of wave regimes on bottom slope, initial landslide depth, its length and width. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献