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A wavepropagation method for conservation laws and balance laws with spatially varying flux functions
 SIAM J. Sci. Comput
, 2002
"... Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a ge ..."
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Cited by 67 (7 self)
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Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A highresolution wavepropagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)− fi−1(Qi−1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be secondorder accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f(q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasisteady problems close to steady state. Key words. finitevolume methods, highresolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X
Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation
 J. Comput. Phys
, 2008
"... We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Ty ..."
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Cited by 20 (2 self)
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We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is wellbalanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth nonnegativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque’s wave propagation algorithm [25] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling. Key words: shallow water equations, hyperbolic conservation laws, finite volume methods, Godunov methods, Riemann solvers, wave propagation, shock capturing methods, tsunami modeling
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 15 (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.
A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes
 SIAM J. Sci. Comput
"... Abstract. The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. One of the main interest of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFR ..."
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Cited by 15 (5 self)
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Abstract. The VFRoe scheme has been recently introduced to approximate the solutions of the shallow water equations. One of the main interest of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in a good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by JinXin [Comm. Pure Appl. Math., 45, 235–276 (1995)], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established and thus the nonnegativity of the water height, obtained involving the VFRoe approach, is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order MUSCL extensions are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
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A ROETYPE SCHEME FOR TWOPHASE SHALLOW GRANULAR FLOWS OVER VARIABLE TOPOGRAPHY
 ESAIM: MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
, 2008
"... We study a depthaveraged model of gravitydriven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. ..."
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Cited by 12 (3 self)
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We study a depthaveraged model of gravitydriven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and nonconservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the onedimensional model equations by a highresolution finite volume scheme based on a Roetype Riemann solver. Wellbalancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over nonflat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.
A Reduced Stability Condition for Nonlinear Relaxation to Conservation Laws
 J. Hyperbolic Differ. Equ
, 2003
"... We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore, Liu in Comm. Pure Appl. Math. 47, 787830, namely the existence of an entr ..."
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Cited by 11 (0 self)
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We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore, Liu in Comm. Pure Appl. Math. 47, 787830, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the ChapmanEnskog expansion. This reduced stability condition has the advantage to involve only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition.
Godunovtype schemes for hyperbolic systems with parameter dependent source. The case of Euler system with friction
, 2009
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Finite Volume Methods for Nonlinear Elasticity in Heterogeneous Media
, 2001
"... An approximate Riemann solver is developed for the equations of nonlinear elasticity in a heterogeneous medium, where each grid cell has an associated density and stressstrain relation. The nonlinear flux function is spatially varying and a wave decomposition of the flux difference across a cell in ..."
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Cited by 8 (3 self)
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An approximate Riemann solver is developed for the equations of nonlinear elasticity in a heterogeneous medium, where each grid cell has an associated density and stressstrain relation. The nonlinear flux function is spatially varying and a wave decomposition of the flux difference across a cell interface is used to approximate the wave structure of the Riemann solution. This solver is used in conjunction with a highresolution finitevolume method using the CLAWPACK software. As a test problem, elastic waves in a periodic layered medium are studied. Dispersive effects from the heterogeneity, combined with the nonlinearity, lead to solitary wave solutions that are well captured by the numerical method.
A Riemann solver for singlephase and twophase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers
 JOURNAL OF COMPUTATIONAL PHYSICS
, 2010
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