to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Stochastic equations in infinite dimensions

This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questions and outlines mathematical progress in the study of the Keller-Segel model since the first presentation of the equations in 1970.

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From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations become independent of the underlying grid implementation. Modern C++ template techniques are used to provide an interface implementation without big performance losses. The implementation is realized as part of the software environment DUNE (http://dune-project.org/). Numerical tests demonstrate the flexibility and the efficiency of our approach.

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A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE

We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987b). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.

Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems

We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].

Fully Discrete, Entropy Conservative Schemes of Arbitrary Order

We are concerned with the approximation of undercompressive, regularization-sensitive, nonclassical solutions of hyperbolic systems of conservation laws by high-order accurate, conservative, and semidiscrete finite difference schemes. Nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance. Particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables.
First, we investigate the existence and the properties of entropy conservative schemes---a notion due to Tadmor [ Math. Comp., 49 (1987), pp. 91--103]. In particular we exhibit a new five-point scheme which is third-order accurate, at least.
Second, we study a class of entropy stable and high-order accurate schemes satisfying a single cell entropy inequality. They are built from any high-order entropy conservative scheme by adding to it a mesh-independent, numerical viscosity, which preserves the order of accuracy of the base scheme. These schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality. These entropy stable schemes exhibit mild oscillations near shocks and, interestingly, may converge to classical or nonclassical entropy solutions, depending on the sign of their dispersion coefficient.
Then, based on a third-order, entropy conservative scheme, we propose a general scheme for the numerical computation of nonclassical shocks. Importantly, our scheme satisfies a cell entropy inequality. Following Hayes and LeFloch [SIAM J. Numer. Anal., 35 (1998), pp. 2169--2194], we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws. Our results compare favorably with previous analytical and numerical results.
Finally, we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete, Lax--Wendroff type schemes.

High-Order Schemes, Entropy Inequalities, and Nonclassical Shocks

An Introduction to Kinetic Schemes for Gas Dynamics.- An Introduction to Nonclassical Shocks of Systems of Conservation Laws.- Viscosity and Relaxation Approximation for Hyperbolic Systems of Conservation Laws.- A Posteriori Error Analysis and Adaptivity for Finite Element Approximations of Hyperbolic Problems.- Numerical Methods for Gasdynamic Systems on Unstructured Meshes.

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An Introduction to Recent Developments in Theory and Numerics for Conservation Laws

We consider a compressible fluid that can occur in a liquid and in a vapour phase. By means of a variational method and basic thermodynamical laws evolution equations to describe the isothermal dynamics can be derived. This method requires only the knowledge of the free energy. Starting from the Van-der-Waals free energy one obtains the classical Navier-Stokes-Korteweg system. However the Van-der-Waals free energy is not the only possible choice. Non-local energies might be physically more relevant. In this paper we consider this class of energies and introduce a new non-local variant of the classical Navier-Stokes-Korteweg system. To ensure the wellposedness of the non-local system we present a short-time existence theorem for the Cauchy-problem.

On local and non‐local Navier‐Stokes‐Korteweg systems for liquid‐vapour phase transitions

Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.

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Christian Rohde

Papers

Fully Discrete, Entropy Conservative Schemes of Arbitrary Order

High-Order Schemes, Entropy Inequalities, and Nonclassical Shocks

An Introduction to Recent Developments in Theory and Numerics for Conservation Laws

On local and non‐local Navier‐Stokes‐Korteweg systems for liquid‐vapour phase transitions

The sharp-interface approach for fluids with phase change: riemann problems and ghost fluid techniques