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62
High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallowwater systems
 Math. Comp
"... Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO r ..."
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Abstract. This paper is concerned with the development of high order methods for the numerical approximation of onedimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the wellbalanced properties of the resulting schemes. Finally, we will focus on applications to shallowwater systems. 1.
On the Computation of Roll Waves
 Math. Model. Num. Anal
, 2000
"... incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the nume ..."
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Cited by 5 (2 self)
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incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the numerical approximation of this problem is its linear instability. Numerical roundoff error can easily overtake the numerical solution and yields false roll wave solution at the steady state.
A SUBSONICWELLBALANCED RECONSTRUCTION SCHEME FOR SHALLOW WATER FLOWS
"... Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reco ..."
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Cited by 5 (3 self)
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Abstract. We consider the SaintVenant system for shallow water flows with nonflat bottom. In the past years, efficient wellbalanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steadystate reconstruction that allows to derive a subsonicwellbalanced scheme, preserving exactly all the subsonic steady states. It generalizes the now wellknown hydrostatic solver, and as the latter it preserves nonnegativity of water height and satisfies a semidiscrete entropy inequality. An application to the EulerPoisson system is proposed. 1.
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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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.
Upwinding Sources at Interfaces in Conservation Laws
, 2003
"... Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical ows because of the gravity, and their numerical approximation leads to speci c diculties. In the context of nite volume schemes, many authors have proposed to Upwind Sources at Interfac ..."
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Cited by 3 (0 self)
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Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical ows because of the gravity, and their numerical approximation leads to speci c diculties. In the context of nite volume schemes, many authors have proposed to Upwind Sources at Interfaces, i.e. the \U. S. I." method, while a cellcentered treatment seems more natural. This note gives a general mathematical formalism for such schemes. We de ne consistency and give a stability condition for the \U. S. I." method. We relate the notion of consistency to the \wellbalanced" property, but its stability remains open, and we also study second order approximations as well as error estimates. The general case of a nonuniform spatial mesh is particularly interesting, motivated by two dimensional problems set on unstructured grids.
Upwinding of source term at interface for Euler equations with high friction, in "Computers and Mathematics with Applications
, 2006
"... with high friction ..."
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.
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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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
Asymptotic Highorder schemes for integrodifferential problems arising in markets with jumps
, 2006
"... In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore ..."
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In this paper we deal with the numerical approximation of integrodifferential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are highorder accurate for large time regimes. Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic highorder schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.