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33
A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows
 SIAM J. Sci. Comput
"... Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when com ..."
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Cited by 44 (4 self)
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Abstract. We consider the SaintVenant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the socalled wellbalanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a wellbalanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast wellbalanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.
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 31 (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
Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review
, 2012
"... Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schem ..."
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Cited by 10 (5 self)
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Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptoticpreserving (AP) schemes are schemes that are efficient in these asymptotic regimes. The designing principle of AP schemes is to preserve, at the discrete level, the asymptotic limit that drives one (usually the microscopic) equation to its asymptotic (macroscopic) equation. An AP scheme is based on solving the microscopic equation, instead of using a multiphysics approach that couples different physical laws at different scales. When the small scale is not numerically resolved, an AP scheme automatically becomes a macroscopic solver for the limiting equation. The AP methodology offers simple, robust and efficient computational methods for a large class of multiscale kinetic, hyperbolic and other physical problems. This
An asymptotic high order masspreserving scheme for a hyperbolic model of chemotaxis
 SIAM J. Num. Anal
"... Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutio ..."
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Cited by 8 (2 self)
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Abstract. We introduce a new class of finite difference schemes for approximating the solutions to an initialboundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.
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 6 (3 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.
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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Cited by 6 (2 self)
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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.
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.
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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Cited by 3 (2 self)
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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.
The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous crosssection
 M2AN MATH. MODEL. NUMER. ANAL
, 2008
"... We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch, and Murat. Observing that th ..."
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Cited by 2 (0 self)
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We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch, and Murat. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system.
HighResolution Finite Volume Methods for Dusty Gas, Jets and Plumes”, submitted to
 SIAM J. Sci. Comput
"... Abstract. We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A highreso ..."
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Cited by 2 (1 self)
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Abstract. We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A highresolution wavepropagation algorithm is used to solve the equations numerically. The onedimensional algorithm is shown to give agreement with a shock tube test problem in the literature. The twodimensional algorithm has been applied to model expolsive volcanic eruptions in which an axisymmetric jet of hot dusty gas is injected into the atmosphere and the expected behavior is observed at two different vent velocities. The methodology described here, with extensions to three dimensions and adaptive mesh refinement, is being used for more detailed studies of volcanic jet processes. Key words. Finite volume methods, highresolution methods, volcanic flows, dusty gas, plumes, jets, shocks