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111
Wave Propagation Methods for Conservation Laws with Source Terms
 In preparation
, 1998
"... . An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm ..."
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Cited by 19 (3 self)
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. An inhomogeneous system of conservation laws will exhibit steady solutions when flux gradients are balanced by source terms. These steady solutions are difficult for many numerical methods (e.g., fractional step methods) to capture and maintain. Recently, a quasisteady wavepropagation algorithm was developed and used to compute nearsteady shallow water flow over variable topography. In this paper we extend this algorithm to nearsteady flow of an ideal gas subject to a static gravitational field. The method is implemented in the software package clawpack. The ability of this method to capture perturbed quasisteady solutions is demonstrated with numerical examples. 1. Introduction We consider the Euler equations in conservation form @ t q +r \Delta f (q) = / (q) (1) where q 2 R m is a vector of conserved quantities, f : R m ! R m is the flux, and / is a source term due to a static gravitational field. It is well known that if f is a nonlinear function of q as for the Eule...
Diffusion Limit Of The Lorentz Model: Asymptotic Preserving Schemes
"... This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right ..."
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Cited by 19 (2 self)
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This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right discrete diusion equation with the same value of the diusion coecient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.
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 (7 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 14 (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
DassFlow v1.0: a variational data assimilation software for 2D river flows
, 2007
"... apport de recherche ISSN 02496399 ISRN INRIA/RR6150FR+ENGDassFlow v1.0: a variational data assimilation software for 2D river flows ..."
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Cited by 14 (7 self)
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apport de recherche ISSN 02496399 ISRN INRIA/RR6150FR+ENGDassFlow v1.0: a variational data assimilation software for 2D river flows
A Wave Propagation Algorithm for Hyperbolic Systems on Curved Manifolds
"... An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, sh ..."
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Cited by 13 (0 self)
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An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and highresolution shockcapturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary onedimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.
An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, Special issue dedicated to the 70th birthday of Professor ZhongCi Shi
 J. Comput. Math
, 2004
"... Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday 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 w ..."
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Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday 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 onedimensional nozzle flows. We use the interface value, rather than the cellaverages, for the source terms, which results in a wellbalanced 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 supercritical 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.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
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How to solve systems of conservation laws numerically using the graphics processor as a highperformance computational engine
 Quak (Eds.), Geometric Modelling, Numerical Simulation, and Optimization: Industrial Mathematics at SINTEF
, 2005
"... Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how the ..."
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Cited by 9 (2 self)
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Summary. The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax–Friedrichs scheme and the Lax–Wendroff scheme. Using a simple example, we show how these two schemes fail to give accurate approximations to solutions containing discontinuities. We then introduce a general class of semidiscrete finitevolume schemes that are designed to produce accurate resolution of both smooth and nonsmooth parts of the solution. Using this special class we wish to introduce the reader to the basic principles used to design modern highresolution schemes. As examples of systems of conservation laws, we consider the shallowwater equations for water waves and the Euler equations for the dynamics of an ideal gas. The second theme in the paper is how programmable graphics processor units (GPUs or graphics cards) can be used to efficiently compute numerical solutions of these systems. In contrast to instruction driven microprocessors (CPUs), GPUs subscribe to the datastreambased computing paradigm and have been optimised for high throughput of large data streams. Most modern numerical methods for hyperbolic conservation laws are explicit schemes defined over a grid, in which the unknowns at each grid point or in each grid cell can be updated independently of the others. Therefore such methods are particularly attractive for implementation using datastreambased processing. 1
FINITE VOLUME METHODS AND ADAPTIVE REFINEMENT FOR GLOBAL TSUNAMI PROPAGATION AND LOCAL INUNDATION.
"... The shallow water equations are a commonly accepted approximation governing tsunami propagation. Numerically capturing certain features of local tsunami inundation requires solving these equations in their physically relevant conservative form, as integral conservation laws for depth and momentum. T ..."
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Cited by 9 (2 self)
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The shallow water equations are a commonly accepted approximation governing tsunami propagation. Numerically capturing certain features of local tsunami inundation requires solving these equations in their physically relevant conservative form, as integral conservation laws for depth and momentum. This form of the equations presents challenges when trying to numerically model global tsunami propagation, so often the best numerical methods for the local inundation regime are not suitable for the global propagation regime. The different regimes of tsunami flow belong to different spatial scales as well, and require correspondingly different grid resolutions. The long wavelength of deep ocean tsunamis requires a large global scale computing domain, yet near the shore the propagating energy is compressed and focused by bathymetry in unpredictable ways. This can lead to large variations in energy and runup even over small localized regions. We have developed a finite volume method to deal with the diverse flow regimes of tsunamis. These methods are well suited for the inundation regime—they are robust in the presence of bores and steep gradients, or drying regions, and can capture the inundating shoreline and runup features. Additionally, these methods are wellbalanced, meaning that they can appropriately model global propagation. To deal with the disparate spatial scales, we have used adaptive refinement algorithms originally developed for gas dynamics, where often steep variation is highly localized at a given time, but moves throughout the domain. These algorithms allow evolving Cartesian subgrids that can move with the propagating waves and highly resolve local inundation of impacted areas in a single global scale computation. Because the dry regions are part of the computing domain, simple rectangular cartesian grids eliminate the need for complex shorelinefitted mesh generation. Science of Tsunami Hazards, Vol. 24, No. 5, page 319 (2006)