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CONVERGENCE OF GODUNOV TYPE METHODS FOR A CONSERVATION LAW WITH A SPATIALLY VARYING DISCONTINUOUS FLUX FUNCTION
"... Abstract. We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these clas ..."
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Abstract. We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a twostep approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in L1. This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in twophase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems. 1.
Finite volume scheme for twophase flow in heterogeneous porous media involving capillary pressure discontinuities
 M2AN Math. Model. Numer. Anal
"... We study a onedimensional model for twophase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission condi ..."
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We study a onedimensional model for twophase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existenceuniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model. MSC subject classification. 35R05, 65M12
ON VANISHING VISCOSITY APPROXIMATION OF CONSERVATION LAWS WITH DISCONTINUOUS FLUX.
, 2009
"... This note is devoted to a characterization of the vanishing viscosity limit for multidimensional conservation laws of the form ut + div f(x, u) = 0, ut=0 = u0 in the domain R + × R N. The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the spac ..."
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This note is devoted to a characterization of the vanishing viscosity limit for multidimensional conservation laws of the form ut + div f(x, u) = 0, ut=0 = u0 in the domain R + × R N. The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities of f(·, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of R N. We define “GV Ventropy solutions ” (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the L 1 contraction principle for the GV Ventropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation u ε t + div (f(x, u ε)) = ε∆u ε, u ε t=0 = u0, ε ↓ 0, of the conservation law. We show that, provided u ε, ε> 0, enjoy a uniform L ∞ bound and the flux f(x, ·) is nondegenerately nonlinear, vanishing viscosity
The l 1 error estimates for a Hamiltonianpreserving scheme for the Liouville equation with piecewise constant potentials
 SIAM J. Num. Anal
, 2008
"... Abstract. We study the l 1error of a Hamiltonianpreserving scheme, developed in [11], for the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through ..."
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Abstract. We study the l 1error of a Hamiltonianpreserving scheme, developed in [11], for the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through barriers, and of the high frequency waves through interfaces. We use the l 1error estimates established in [30, 28] for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients. We prove that the scheme with the Dirichlet incoming boundary conditions is l 1convergent for a class of bounded initial data, and derive the onehalfth order l 1error bounds with explicit coefficients. The initial conditions can be satisfied by applying the decomposition technique proposed in [10] for solving the Liouville equation with measurevalued initial data, which arises in the semiclassical limit of the linear Schrödinger equation. Key words. Liouville equations, Hamiltonian preserving schemes, piecewise constant potentials, error estimate, half order error bound, semiclassical limit AMS subject classifications. 65M06, 65M12, 65M25, 35L45, 70H99 1. Introduction. In [11]
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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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.
VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
"... Abstract. We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L ∞ contractive semigroup. We defin ..."
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Abstract. We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L ∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to “internal boundaries”. The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by “viscosity solution ” we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.
On the upstream mobility scheme for twophase flow in porous media
 Comput. Geosci
"... apport de recherche ISSN 02496399 ISRN INRIA/RR6789FR+ENGinria00353627, version 1 15 Jan 2009On the upstream mobility scheme for twophase flow in porous media ..."
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apport de recherche ISSN 02496399 ISRN INRIA/RR6789FR+ENGinria00353627, version 1 15 Jan 2009On the upstream mobility scheme for twophase flow in porous media
Applications of the DFLU flux to systems of conservation laws
"... apport de recherche ISSN 02496399 ISRN INRIA/RR7009FR+ENGApplications of the DFLU flux to systems of conservation laws ..."
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apport de recherche ISSN 02496399 ISRN INRIA/RR7009FR+ENGApplications of the DFLU flux to systems of conservation laws
A note on an extended clarifierthickener model with singular source and sink terms
 SCIENTIA SERIES A: MATHEMATICAL SCIENCES, VOL. 13 (2006), 36–44
, 2006
"... A onedimensional clarifierthickener (CT) model, which includes a singular feed source, can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. The CT model is extended herein by a singular sink through which material is extracted. The sink gi ..."
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A onedimensional clarifierthickener (CT) model, which includes a singular feed source, can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. The CT model is extended herein by a singular sink through which material is extracted. The sink gives rise to a new nonconservative transport term. The paper summarizes a recent wellposedness analysis and presents a numerical method for the extended model. A numerical example is presented.