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19
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 28 (5 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
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 7 (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.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
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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 7 (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.
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 4 (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.
High Resolution Methods and Adaptive Refinement for Tsunami Propagation and Inundation.
"... We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents ..."
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Cited by 4 (1 self)
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We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunovtype methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents some unique challenges that must be overcome. We describe some of the specific difficulties associated with tsunami modeling, and summarize some numerical approaches that we have used to overcome these challenges. For instance, we have developed a wellbalanced Riemann solver that is appropriate in the deep ocean regime as well as robust in nearshore and dry regions. Additionally, we have extended adaptive refinement algorithms to this application. We briefly describe some of the modifications necessary for using these adaptive methods for tsunami modeling.
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 3 (1 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
Numerical modeling of twophase gravitational granular flows with bottom topography
 Serre (Eds.), Hyperbolic Problems: Theory, Numerics and Applications, Proceedings of the Eleventh International Conference on Hyperbolic Problems
, 2008
"... Summary. We study a depthaveraged model of gravitydriven mixtures of solid grains and fluid moving over variable basal surface. The particular application we are interested in is the numerical description of geophysical flows such as avalanches and debris flows, which typically contain both solid ..."
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Cited by 2 (1 self)
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Summary. We study a depthaveraged model of gravitydriven mixtures of solid grains and fluid moving over variable basal surface. The particular application we are interested in is the numerical description of geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The depthaveraged mass and momentum equations for the solid and fluid components form a nonconservative system, where nonconservative terms involving the derivatives of the unknowns couple together the sets of equations of the two phases. The system can be shown to be hyperbolic at least when the difference of velocities of the two constituents is sufficiently small. We numerically solve the model equations in one dimension by a 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. 1
A Class of Residual Distribution Schemes and Their Relation to Relaxation Systems
, 711
"... Residual distributions (RD) schemes are a class of of highresolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by fini ..."
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Cited by 1 (0 self)
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Residual distributions (RD) schemes are a class of of highresolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [J. Comp. Phys. 172, 572 (2001)] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, LaxFriedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)]. In this work we extend LeVeque and Pelanti’s results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard Nscheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local LaxFriedrichs scheme. Once this new scheme is established, we test it on a standard test case involving steadystate computations of the 2D Euler equations around the NACA 0012 airfoil. We show that through the use of linearpreserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the Nscheme, despite being computationally less expensive. Key words: residual distribution schemes, relaxation systems, approximate Riemann solvers, finite volume methods, hyperbolic conservation laws