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13
Equilibrium schemes for scalar conservation laws with stiff sources
 Math. Comp
, 2003
"... Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory resu ..."
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Cited by 38 (4 self)
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Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a socalled equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients. 1.
Asymptoticpreserving & wellbalanced schemes for radiative transfer and the Rosseland approximation
, 2003
"... We are concerned with efficient numerical simulation of the radiative transfer equations... ..."
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Cited by 18 (2 self)
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We are concerned with efficient numerical simulation of the radiative transfer equations...
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
"... ..."
TIME–SPLITTING SCHEMES AND MEASURE SOURCE TERMS For A Quasilinear Relaxing System
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2003
"... Several singular limits are investigated in the context of a 2×2 system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a L² projection operator are concentrated on a discrete lattice by means of Dirac measures. Thi ..."
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Cited by 4 (4 self)
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Several singular limits are investigated in the context of a 2×2 system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a L² projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some timesplitting numerical schemes.
THE RIEMANN PROBLEM FOR THE SHALLOW WATER EQUATIONS WITH DISCONTINUOUS TOPOGRAPHY
, 2007
"... We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves ..."
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Cited by 2 (2 self)
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We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is nonstrictly hyperbolic and does not admit a fully conservative form, and we establish the existence of twoparameter wave sets, rather than wave curves. The selection of admissible waves is particularly challenging. Our construction is fully explicit, and leads to formulas that can be implemented numerically for the approximation of the general initialvalue problem.
WELLBALANCED SCHEMES FOR CONSERVATION LAWS WITH SOURCE TERMS BASED ON A LOCAL DISCONTINUOUS FLUX FORMULATION
, 2009
"... We propose and analyze a finite volume scheme of the Godunov type for conservation laws with source terms that preserve discrete steady states. The scheme works in the resonant regime as well as for problems with discontinuous flux. Moreover, an additional modification of the scheme is not required ..."
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Cited by 2 (0 self)
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We propose and analyze a finite volume scheme of the Godunov type for conservation laws with source terms that preserve discrete steady states. The scheme works in the resonant regime as well as for problems with discontinuous flux. Moreover, an additional modification of the scheme is not required to resolve transients, and solutions of nonlinear algebraic equations are not involved. Our wellbalanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux. Due to the difficulty of obtaining BV estimates, we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero. We include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature.
First and second order error estimates for the Upwind Source at Interface method *
"... Abstract The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced in [29] is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove Lperror estimates, 1< = p ..."
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Abstract The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced in [29] is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove Lperror estimates, 1< = p <+1, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem [6]. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question to derive second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the U.S.I. method. 1 Introduction We consider the initial value problem for a transport equation with nonlinearsource term, in one space dimension, @tu + @xu = B(x, u), t 2 R+, x 2 R, (1.1) u(0, x) = u0(x) 2 Lp(R) " L1(R), 1 < = p < +1, (1.2)
A Robust and EntropySatisfying Numerical Scheme for Fluid Flows in Discontinuous Nozzles
, 2013
"... We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the crosssection and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the us ..."
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Cited by 1 (1 self)
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We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the crosssection and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous with arbitrary large jumps. To do so, we introduce in the first step of the relaxation solver a singular dissipation measure superposed on the standing wave which enables us to control the approximate speeds of sound and thus, the time step, even for extreme initial data. Keywords: Discontinuous nozzle flows, relaxation techniques, Riemann problem. AMS subject classifications: 76S05, 35L60, 35F55. 1