Results 11  20
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35
On Strongly Degenerate ConvectionDiffusion Problems Modeling SedimentationConsolidation Processes
, 1999
"... . We investigate initialboundary value problems for a quasilinear strongly degenerate convectiondiffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentationconsolidation processes. Existence of entropy solutions belongi ..."
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Cited by 15 (10 self)
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. We investigate initialboundary value problems for a quasilinear strongly degenerate convectiondiffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentationconsolidation processes. Existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second order equations of Kruzkov's method of "doubling of the variables". The application to a sedimentationconsolidation model is illustrated by two numerical examples. 1. Introduction In this paper, we consider quasilinear strongly degenerate parabolic equations of the type @ t u + @ x (q(t)u + f(u)) = @ 2 x A(u); (x; t) 2 Q T ; A(u) := Z u 0 a(s) ds; a(u) 0; (1.1) where QT :=\Omega \Theta T,\Omega := (0;...
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 14 (3 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.
The Corrected Operator Splitting Approach Applied To A Nonlinear AdvectionDiffusion Problem
 COMPUT. METHODS APPL. MECH. ENGRG
, 1997
"... Socalled corrected operator splitting methods are applied to a 1D scalar advectiondiffusion equation of BuckleyLeverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piecewise linear finite element ..."
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Cited by 13 (10 self)
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Socalled corrected operator splitting methods are applied to a 1D scalar advectiondiffusion equation of BuckleyLeverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piecewise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, we extend the corrected operator splitting to 2D equations by means of dimensional splitting, and we apply it to a BuckleyLeverett type problem including gravitational effects.
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 13 (4 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 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.
A Continuous Dependence Result For Nonlinear Degenerate Parabolic Equations With Spatially Dependent Flux Function
 Proc. Hyp
, 2000
"... . We study entropy solutions of nonlinear degenerate parabolic equations of form u t + div k(x)f(u) = A(u), where k(x) is a vectorvalued function and f(u); A(u) are scalar functions. We prove a result concerning the continuous dependence on the initial data, the ux function k(x)f(u), and the di ..."
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Cited by 12 (5 self)
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. We study entropy solutions of nonlinear degenerate parabolic equations of form u t + div k(x)f(u) = A(u), where k(x) is a vectorvalued function and f(u); A(u) are scalar functions. We prove a result concerning the continuous dependence on the initial data, the ux function k(x)f(u), and the diusion function A(u). This paper complements previous work [7] by two of the authors, which contained a continuous dependence result concerning the initial data and the ux function k(x)f(u). 1. Introduction In this paper we are concerned with entropy solutions of the initial value problem (1.1) u t + div k(x)f(u) = A(u); u(x; 0) = u 0 (x) for (x; t) 2 T = R d (0; T ) with T > xed. In (1.1), u(x; t) is the scalar unknown function that is sought, k(x)f(u) is the ux function, and A = A(u) is the diusion function. We always assume that k : R d ! R, f : R ! R, and A : R ! R satisfy (1.2) ( k 2 W 1;1 loc (R d ); k; divk 2 L 1 (R d ); f 2 Lip loc (R); f(0) = 0; A 2 Li...
A Note On The Uniqueness Of Entropy Solutions Of Nonlinear Degenerate Parabolic Equations
 J. Math. Anal. Appl
, 2001
"... . Following the lead of Carrillo [6], recently several authors have used Kruzkov's device of \doubling the variables" to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order dierential operator is not allowed to depend exp ..."
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Cited by 11 (5 self)
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. Following the lead of Carrillo [6], recently several authors have used Kruzkov's device of \doubling the variables" to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations. In all these results, the second order dierential operator is not allowed to depend explicitly on the spatial variable, which certainly restricts the range of applications of entropy solution theory. The purpose of this paper is to extend a version of Carrillo's uniqueness result found in Karlsen and Risebro [14] to a class of degenerate parabolic equations with spatially dependent second order dierential operator. The class is large enough to encompass several interesting nonlinear partial dierential equations coming from the theory of porous media ow and the phenomenological theory of sedimentationconsolidation processes. 1.
L 1 framework for continuous dependence and error estimates for quasilinear degenerate parabolic equations
, 2002
"... Abstract. We develop a general L1 –framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the ChenPerthame kinetic approach [9]. We apply our L1 –framework to establish an explicit estimate for continuous dependenc ..."
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Cited by 10 (4 self)
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Abstract. We develop a general L1 –framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the ChenPerthame kinetic approach [9]. We apply our L1 –framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convectiondiffusion model equation and derive an L1 error estimate for an upwindcentral finite difference scheme.
Front tracking and operator splitting for nonlinear degenerate convectiondiffusion equations
 Parallel Solution of Partial Dierential Equations, volume 120 in the IMA Volumes in Mathematics and its Applications
, 1997
"... We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convectiondiffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The ..."
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Cited by 9 (6 self)
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We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convectiondiffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.
Renormalized Entropy Solutions For Quasilinear Anisotropic Degenerate Parabolic Equations
 SIAM J. MATH. ANAL
, 2003
"... We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L¹ data. This paper complements the work by Chen and Perthame [19], who developed a pure L¹ theory based on the notion of kin ..."
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Cited by 9 (5 self)
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We prove the well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L¹ data. This paper complements the work by Chen and Perthame [19], who developed a pure L¹ theory based on the notion of kinetic solutions.