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26
Why many theories of shock waves are necessary. Convergence error in formally pathconsistent schemes
, 2008
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A new approach of high order wellbalanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms
 Communications in Computational Physics
"... Abstract. Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order wellbalanced schemes to a class of hyperbolic systems with separable source terms. In this paper, w ..."
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Cited by 16 (5 self)
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Abstract. Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order wellbalanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order wellbalanced finite volume weighted essentially nonoscillatory (WENO) schemes and RungeKutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the nonoscillatory property for general solutions with discontinuities, and the genuine high
2008: A discontinuous Galerkin finite element model for morphological evolution under shallow flows
"... We present a discontinuous Galerkin finite element method for two depthaveraged twophase flow models. One of these models contains nonconservative products for which we developed a discontinuous Galerkin finite element formulation in Rhebergen et al. (2008) J. Comput. Phys. 227, 18871922. The ot ..."
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Cited by 13 (2 self)
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We present a discontinuous Galerkin finite element method for two depthaveraged twophase flow models. One of these models contains nonconservative products for which we developed a discontinuous Galerkin finite element formulation in Rhebergen et al. (2008) J. Comput. Phys. 227, 18871922. The other model is a new depthaveraged twophase flow model we introduce for shallow twophase flows that does not contain nonconservative products. We will compare numerical results of both models and qualitatively validate the models against a laboratory experiment. Furthermore, because of spurious oscillations that may occur near discontinuities, a WENO slope limiter is applied in conjunction with a discontinuity detector to detect regions where spurious oscillations appear. Key words: discontinuous Galerkin finite element methods, multiphase flows, nonconservative products, slope limiter, discontinuity detector
Highorder finite volume WENO schemes for the shallow water equations with dry states
, 2011
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Wellbalanced and energy stable schemes for the shallow water equations with discontinuous topography
 J. Comput. Phys
"... Abstract. We consider the shallow water equations with nonflat bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states of lake at rest as well as moving equilibrium states. We desig ..."
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Abstract. We consider the shallow water equations with nonflat bottom topography. The smooth solutions of these equations are energy conservative, whereas weak solutions are energy stable. The equations possess interesting steady states of lake at rest as well as moving equilibrium states. We design energy conservative finite volume schemes which preserve (i) the lake at rest steady state in both one and two space dimensions, and (ii) onedimensional moving equilibrium states. Suitable energy stable numerical diffusion operators, based on energy and equilibrium variables, are designed to preserve these two types of steady states. Several numerical experiments illustrating the robustness of the energy preserving and energy stable wellbalanced schemes are presented.
GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations
"... We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of ts ..."
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We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multirate AdamsBashforth scheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. We compare the performance of the kernels expressed in a portable threading language OCCA, when cross compiled with OpenCL, CUDA, and OpenMP at runtime.
A FULLY WELLBALANCED, POSITIVE AND ENTROPYSATISFYING GODUNOVTYPE METHOD FOR THE SHALLOWWATER EQUATIONS
"... AMS subject classifications. 65M60, 65M12 Key words. Shallowwater equations, steady states, finite volume schemes, wellbalanced property, positive preserving scheme, entropy preserving scheme Abstract. This work is devoted to the derivation of a fully wellbalanced numerical scheme for the wellk ..."
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AMS subject classifications. 65M60, 65M12 Key words. Shallowwater equations, steady states, finite volume schemes, wellbalanced property, positive preserving scheme, entropy preserving scheme Abstract. This work is devoted to the derivation of a fully wellbalanced numerical scheme for the wellknown shallowwater model. During the last two decades, several wellbalanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the socalled lake at rest. By fully wellbalanced, we mean here that the proposed Godunovtype method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality. 1. Introduction. The
CentralUpwind Scheme on Triangular Grids for the SaintVenant System of Shallow Water Equations
"... Abstract. We consider a novel secondorder centralupwind scheme for the SaintVenant system of shallow water equations on triangular grids which was originally introduced in [3]. Here, in several numerical experiments we demonstrate accuracy, high resolution and robustness of the proposed method. ..."
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Abstract. We consider a novel secondorder centralupwind scheme for the SaintVenant system of shallow water equations on triangular grids which was originally introduced in [3]. Here, in several numerical experiments we demonstrate accuracy, high resolution and robustness of the proposed method.