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Handbook of Numerical Analysis
"... A consistent intermediate wave speed for a wellbalanced HLLC solver ..."
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A consistent intermediate wave speed for a wellbalanced HLLC solver
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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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
Quadtreeadaptive tsunami modelling
 Ocean Dynamics
, 2011
"... The wellbalanced, positivitypreserving scheme of Audusse et al, 2004, for the solution of the SaintVenant equations with wetting and drying, is generalised to an adaptive quadtree spatial discretisation. The scheme is validated using an analytical solution for the oscillation of a fluid in a para ..."
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The wellbalanced, positivitypreserving scheme of Audusse et al, 2004, for the solution of the SaintVenant equations with wetting and drying, is generalised to an adaptive quadtree spatial discretisation. The scheme is validated using an analytical solution for the oscillation of a fluid in a parabolic container, as well as the classic Monai tsunami laboratory benchmark. An efficient database system able to dynamically reconstruct a multiscale bathymetry based on extremely large datasets is also described. This combination of methods is sucessfully applied to the adaptive modelling of the 2004 Indian ocean tsunami. Adaptivity is shown to significantly decrease the exponent of the power law describing computational cost as a function of spatial resolution. The new exponent is directly related to the fractal dimension of the geometrical structures characterising tsunami propagation. The implementation of the method as well as the data and scripts necessary to reproduce the results presented are freely available as part of the opensource Gerris Flow Solver framework. 1
A consistent intermediate wave speed for a wellbalanced hllc
, 2008
"... Abstract. In this work, we extend the HLLC solver to nonhomogeneous ShallowWater Equations (SWE) (equations are either 1D SWE with pollutant or 2D SWE, both with topography term). The extension done is based on a modification of the numerical flux which depends on the source term (the topography) ..."
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Abstract. In this work, we extend the HLLC solver to nonhomogeneous ShallowWater Equations (SWE) (equations are either 1D SWE with pollutant or 2D SWE, both with topography term). The extension done is based on a modification of the numerical flux which depends on the source term (the topography) ; it is not based on an upwinding of the source term. We obtain a wellbalanced scheme and a consistent approximation of the intermediate wave speed. Numerical results are presented, and we show that a wellbalanced scheme without a consistent approximation of the intermediate wave speed can produce bad results. Une vitesse intermédiaire consistante pour un schéma HLLC bienéquilibré. Résumé Nous présentons une modification du schéma HLLC pour application auxéquations de SaintVenant non homogènes avec transport de polluants. Cette nouvelle version du schéma est reliéeà la définition d'une vitesse d'onde intermédiaire consistante. Un exemple est donné afin d'illuster comment une mauvaise approximation de la vitesse intermédiaire peut entraîner de mauvais résultats sur l'approximation de la concentration en polluant même avec un schéma bień equilibré. Version française abrégée Nous proposons une extension du schéma HLLC pour leś equations de SaintVenant avec topographie et propagation polluants (voir [6]). Pour ce système le schéma HLLC peut s'écrire en fonction du flux numérique provenant du schéma HLL. Pour le cas homogène, ce schéma est basé sur le fait que la troisième composante du flux peut s'écrire en fonction de la première composante du flux et de la concentration en polluant. La définition de la troisième composante du flux est La définition de ϕ * est simplement la concentration du polluantà gauche ouà droite de l'intervalle x = x i+1/2 en fonction du signe de la vitesse intermédiaire S * . Pour le cas non homogène nous considérons un schéma de type HLLécrit comme fonction du flux numérique qui dépend de la définition du terme de front (voir [4], [8]). Ceci va nous permettre d'obtenir une extension naturelle du schéma HLLC au cas non homogèneà partir du cas homogène. Nous prouvons que le schéma proposé calcule exactement la solution stationnaire de l'eau au repos et qu'il est asymptotiquement bienéquilibré (voir [4]) indépendamment de la définition de S * . Nous proposons alors une modification de S * , qui dépend de la topographie, qui vaut zéro au repos et quiévite des pics anormaux de polluants. En guise d'exemple, nous exhibons un test numérique sur lequel nous observons l'importance de cette modification. L'expression de S * est donnée par (5).
Discontinuous Galerkin Method for 1D Shallow Water Flow with Water Surface Slope Limiter
"... Abstract—A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing RungeKutta discontinuous Galerkin finite ele ..."
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Abstract—A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing RungeKutta discontinuous Galerkin finite element method with slope limiter schemes based on water surface slope and water depth are used to solve onedimensional shallow water equations. For each slope limiter, three different Riemann solvers based on HLL, LF, and Roe flux functions are used. The proposed water surface based slope limiter scheme is easy to implement and shows better conservation property compared to the slope limiter based on water depth. Of the three flux functions, the Roe approximation provides the best results while the LF function proves to be least suitable when used with either slope limiter scheme. Keywords—Discontinuous finite element, TVD RungeKutta scheme, slope limiters, Riemann solvers, shallow water flow. O I.
Front tracking for scalar balance equations
 J. Hyperbolic Differ. Equ
"... Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the f ..."
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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and nonstiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.
Highorder wellbalanced finite difference weno schemes for a class of hyperbolic systems with source terms
 J. Sci. Comput
, 2006
"... In this paper, we generalize the high order wellbalanced finite difference weighted essentially nonoscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable so ..."
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In this paper, we generalize the high order wellbalanced finite difference weighted essentially nonoscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), 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 order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems. KEY WORDS: Hyperbolic balance laws; WENO scheme; highorder accuracy; source term; conservation laws; shallow water equation; elastic wave equation; chemosensitive movement; nozzle flow; two phase flow. 1.
An entropycorrection free solver for non homogeneous shallow water equations. ESAIM: M2AN
, 2003
"... Abstract. In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the nonhomogeneous case, in such a way that all equilibria are compu ..."
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Abstract. In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the nonhomogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.
Numerical Models for Scour and Liquefaction Around Object Under Currents and Waves
, 2008
"... Local scour and liquefaction are two of the most important processes which affect the interactions between fluid, object and sediment when an object (such as bridge pier, offshore foundation, etc.) is exposed to currents and waves. In the present study, numerical models are developed to understand t ..."
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Local scour and liquefaction are two of the most important processes which affect the interactions between fluid, object and sediment when an object (such as bridge pier, offshore foundation, etc.) is exposed to currents and waves. In the present study, numerical models are developed to understand these complicated processes. For the local scour process, twodimensional and threedimensional models are developed respectively. In the twodimensional model, shallow water equations with finite volume method on unstructured mesh are used. The twodimensional model uses the Godunov scheme and approximate Riemann solvers. Hydrodynamics and sediment transport equations are coupled and solved simultaneously. Asymptotic analysis of the system eigenvalues is given and the approximation is compared with the numerical results. The model developed in this thesis can deal with wetting and drying automatically. Discontinuity of the flow, such as a hydraulic jump, can be captured. For the three dimensional model, free water surface and automatic mesh deformation for the bed are incorporated in the model. The Reynolds Averaged NavierStokes (RANS) turbulence model is used to simulate the turbulent flow field. The turbulence model used is k Model. Two interfaces