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71
Convergence of the LaxFriedrichs scheme and stability for conservation laws with a discontinuous spacetime dependent flux
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On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients
, 2009
"... We study nonlinear degenerate parabolic equations where the flux function f(x, t, u) does not depend Lipschitz continuously on the spatial location x. By properly adapting the “doubling of variables” device due to Kruˇzkov [24] and Carrillo [12], we prove a uniqueness result within the class of en ..."
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Cited by 38 (15 self)
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We study nonlinear degenerate parabolic equations where the flux function f(x, t, u) does not depend Lipschitz continuously on the spatial location x. By properly adapting the “doubling of variables” device due to Kruˇzkov [24] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form k(x)f(u), where k(x) is a vectorvalued function and f(u) is a scalar function.
A relaxation scheme for conservation laws with a discontinuous coefficient
 Math. Comp
, 2007
"... Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a wea ..."
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Cited by 38 (8 self)
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Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.
Upwind difference approximations for degenerate parabolic convectiondiffusion equations with a discontinuous coefficient
 IMA J. Numer. Anal
, 2002
"... Abstract. We analyze approximate solutions generated by an upwind difference scheme (of EngquistOsher type) for nonlinear degenerate parabolic convectiondiffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed ..."
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Cited by 31 (9 self)
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Abstract. We analyze approximate solutions generated by an upwind difference scheme (of EngquistOsher type) for nonlinear degenerate parabolic convectiondiffusion equations where the nonlinear convective flux function has a discontinuous coefficient γ(x) and the diffusion function A(u) is allowed to be strongly degenerate (the pure hyperbolic case is included in our setup). The main problem is obtaining a uniform bound on the total variation of the difference approximation u∆, which is a manifestation of resonance. To circumvent this analytical problem, we construct a singular mapping Ψ(γ, ·) such that the total variation of the transformed variable z ∆ = Ψ(γ∆, u∆) can be bounded uniformly in ∆. This establishes strong L1 compactness of z ∆ and, since Ψ(γ, ·) is invertible, also u∆. Our singular mapping is novel in that it incorporates a contribution from the diffusion function A(u). We then show that the limit of a converging sequence of difference approximations is a weak solution as well as satisfying a Kružkovtype entropy inequality. We prove that the diffusion function A(u) is Hölder continuous, implying that the constructed weak solution u is continuous in those regions where the diffusion is nondegenerate. Finally, some numerical experiments are presented and discussed.
Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients
 M2AN Math. Model. Numer. Anal
"... Abstract. We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough ” coefficient function k(x). We show that the EngquistOsher (and hence all monotone) finite difference approximations ..."
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Cited by 25 (12 self)
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Abstract. We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough ” coefficient function k(x). We show that the EngquistOsher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k ′ is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convectiondiffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general L p compactness criterion.
On a Nonlinear Degenerate Parabolic TransportDiffusion Equation . . .
, 2002
"... We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transportdiffusion equation # t u + #x x A(u), A # () where the coefficient #(x) is possibly discontinuous and f(u) is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is p ..."
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Cited by 20 (8 self)
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We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transportdiffusion equation # t u + #x x A(u), A # () where the coefficient #(x) is possibly discontinuous and f(u) is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as # 0 in a suitable sequence of smooth approximations solving the problem above with the transport flux #(x)f() replaced by ## (x)f() and the diffusion function A() replaced by A# (), where ## () is smooth and A # # () > 0. The main technical challenge is to deal with the fact that the total variation u#  uniformly in #, and hence one cannot derive directly strong convergence of . In the purely hyperbolic case (A # 0), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term.
A model of continuous sedimentation of flocculated suspensions in clarifierthickener units
, 2005
"... The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentationconsolidation processes of flocculated suspensions in clarifierthickener units. This model appears in two variants for cylindrical and variable crosssectional area units, respec ..."
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Cited by 19 (5 self)
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The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentationconsolidation processes of flocculated suspensions in clarifierthickener units. This model appears in two variants for cylindrical and variable crosssectional area units, respectively (Models 1 and 2). In both cases, the governing equation is a scalar, strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable. The initial value problem for this equation is analyzed for Model 1. We introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition. A limited analysis of steady states as desired stationary modes of operation is performed. Numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifierthickeners.
A UNIQUENESS CONDITION FOR NONLINEAR CONVECTIONDIFFUSION EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
 JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS
, 2008
"... The paper focuses on the uniqueness issue for scalar convectiondiffusion equations where both the convective flux and diffusion functions have a spatial discontinuity. An interface entropy condition is proposed at such a spatial discontinuity. It implies the Kruˇzkovtype entropy condition presente ..."
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Cited by 12 (1 self)
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The paper focuses on the uniqueness issue for scalar convectiondiffusion equations where both the convective flux and diffusion functions have a spatial discontinuity. An interface entropy condition is proposed at such a spatial discontinuity. It implies the Kruˇzkovtype entropy condition presented by Karlsen et al. [Trans. Royal Norwegian Society Sci. Letters 3, 49 pp, 2003]. They proved uniqueness when the convective flux function satisfies an additional ‘crossing condition’. The crossing condition becomes redundant with the entropy condition proposed here. Thereby, more general flux functions are allowed. Another advantage of the entropy condition is its simple geometrical interpretation, which facilitates the construction of stationary solutions.
On the existence and compactness of a twodimensional resonant system of conservation laws
 Commun. Math. Sci
"... Abstract. We prove the existence of a weak solution to a twodimensional resonant 3×3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with th ..."
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Cited by 12 (3 self)
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Abstract. We prove the existence of a weak solution to a twodimensional resonant 3×3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a twodimensional compensated compactness argument adapted from [E. Tadmor, M. Rascle, and P. Bagnerini, Compensated compactness for 2D conservation laws, J. Hyperbolic Differ. Equ., 2(3), 697712, 2005]. Existence is proved under the assumption that the flux functions in the two directions are linearly independent. Key words. Nonlinear conservation laws, multidimensional, discontinuous fluxes, entropy bounds, weak solutions, existence, compensated compactness AMS subject classifications. 35L65, 35L80 1.
Finite volume schemes for locally constrained conservation laws
, 2009
"... This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin in [CG07]. The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.). We first show t ..."
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Cited by 11 (10 self)
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This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin in [CG07]. The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.). We first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function, as developed by Adimurthi et al. [AMG05] and Bürger et al. [BKT09]. We reformulate accordingly the notion of entropy solution introduced in [CG07], and extend the wellposedness results to the L ∞ framework. Then, starting from a general monotone finite volume scheme for the nonconstrained conservation law, we produce a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process