François James

One-dimensional transport equations with discontinuous coefficients

We consider one-dimensional linear transport equations with bounded but possibly discontinuous coefficient a. The Cauchy problem is studied from two different points of view. In the first case we assume that a is piecewise continuous. We give an existence result and a precise description of the solutions on the lines of discontinuity. In the second case, we assume that a satisfies a one-sided Lipschitz condition. We give existence, uniqueness and general stability results for backward Lipschitz solutions and forward measure solutions, by using a duality method. We prove that the flux associated to these measure solutions is a product by some canonical representative â of a. Key-words. Linear transport equations, discontinuous coefficients, weak stability, duality, product of a measure by a discontinuous function, nonnegative solutions. 1991 Mathematics Subject Classification. Primary 35F10, 35B35, 34A12. To appear in Nonlinear Analysis, TMA ∗Departement de Mathematiques et Applications, UMR CNRS 8553, Ecole Normale Superieure et CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France, francois.bouchut@ens.fr †Mathematiques, Applications et Physique Mathematique d’Orleans, UMR CNRS 6628, Universite d’Orleans, 45067 Orleans Cedex 2, France, james@math.cnrs.fr

Determination of binary competitive equilibrium isotherms from the individual chromatographic band profiles

A numerical solution of the inverse problem of nonlinear chromatography is described and validated. This method allows the determination of best numerical estimates of the coefficients of an isotherm model from the individual elution profiles of the two components of a binary mixture. The sample size must be large enough for the two bands to interfere strongly and for their maximum concentrations to exceed the range within which the isotherm equation is needed. In two cases, excellent agreement was observed between the equilibrium isotherm equations obtained by this new method and those determined by the classical combination of elution by characteristic points and binary frontal analysis. In the first case, the adsorption of the ketoprofen enantiomers on a cellulose-based chiral phase is accounted for by a competitive Bilangmuir isotherm. In the second case, the adsorption of benzyl alcohol and 2-phenylethanol on C18 silica is accounted for by a competitive Langmuir model. The importance of using the proper boundary conditions (i.e., a realistic injection profile) is stressed. The new method seems especially well suited for the rapid determination of the isotherms of enantiomers needed for the computer-assisted optimization of the separation of mixtures of these compounds, e.g., in simulated moving-bed applications.

SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.

Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients

Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

Parameter identification for a model of chromatographic column

Here we are interested in identifying the flux in a system of conservation laws. This problem is ill-posed, but we can deal with the indetermination by using an a priori problem as a constrained minimization of a cost function J. A Lagrangian formulation leads us to a formal computation of that gradient of J. We then perform an exact computation of the gradient of the discrete cost function. Finally, we give a few numerical results obtained through a conjugate gradient method.

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.

Convergence Results for the Flux Identification in a Scalar Conservation Law

Here we study an inverse problem for a quasilinear hyperbolic equation. We start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function. Then we use a Lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation. Despite the fact that the Lagrangian formulation is formal and that the cost function is not necessarily differentiable, a viscous perturbation and a numerical approximation of the problem allow us to justify this computation. When the adjoint problem for the quasi-linear equation admits a smooth solution, then the perturbed adjoint states can be proved to converge to that very solution. The sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdifferential of the cost function. We evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identification of isotherms for conservation laws modeling distillation or chromatography. They are illustrated by numerical examples.

Differentiability with Respect to Initial Data for a Scalar Conservation Law

We linearize a scalar conservation law around an entropy initial datum. The resulting equation is a linear conservation law with discontinuous coefficient, solved in the context of duality solutions, for which existence and uniqueness hold. We interpret these solutions as weak derivatives with respect to the initial data for the nonlinear equation.

Numerical identification of parameters for a model of sedimentation processes

In this paper we present the identification of parameters in the flux and diffusion functions for a quasilinear strongly degenerate parabolic equation which models the physical phenomenon of flocculated sedimentation. We formulate the identification problem as a minimization of a suitable cost function and we derive its formal gradient by means of an adjoint equation which is a backward linear degenerate parabolic equation with discontinuous coefficients. For the numerical approach, we start with the discrete Lagrangian formulation and assuming that the direct problem is discretized by the Engquist–Osher scheme obtain a discrete adjoint state associated with this scheme. The conjugate gradient method allows us to find numerical values of the physical parameters. It also allows us to identify the critical concentration level at which solid flocs begin to touch each other and determine the change of parabolic to hyperbolic behaviour in the model equation.

Convergence results for an inhomogeneous system arising in various high frequency approximations

Summary. This paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of Hamilton-Jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the Schrödinger equation. We first state an existence and uniqueness result in the framework of viscosity and duality solutions. Then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence. As an illustration, some practical computations are provided.