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 vector-valued function and f(u) is a scalar function.

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 Engquist-Osher (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.

. We investigate initial-boundary value problems for a quasilinear strongly degenerate convection-diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modeling of certain sedimentation-consolidation 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 sedimentation-consolidation 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;...

We show how existingmodels for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions formingcompressible sediments (“sedimentation with compression” or “sedimentation-consolidation process”). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov–Tadmor central difference scheme for convection-diffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.

We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion 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.

We design numerical schemes for nonlinear degenerate parabolic systems with possibly dominant convection. These schemes are based on discrete BGK models where both characteristic velocities and the source-term depend singularly on the relaxation parameter. General stability conditions are derived, and convergence is proved to the entropy solutions for scalar equations.

We establish a rate of convergence for a semi-discrete operator splitting method applied to Hamilton-Jacobi equations with source terms. The method is based on sequentially solving a Hamilton-Jacobi equation and an ordinary differential equation. The Hamilton-Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the L 1 error associated with the operator splitting method is bounded by O(t), where t is the splitting (or time) step. This error bound is an improvement over the existing O( p t) bound due to Souganidis [40]. In the one dimensional case, we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous Hamilton-Jacobi equations. It is proved that this fully discrete splitting method possesses a linear convergence rate. Moreover, numerical results are presented to illustrate the theoretical convergence results.

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 Chen-Perthame 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 convection-diffusion model equation and derive an L1 error estimate for an upwind-central finite difference scheme.

. Many numerical methods for (one-dimensional) systems of convection-diffusion equations are based upon an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous s...

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.