1.  Scientific motivation for this project

1.1 Introduction

 Fluid Mechanics is a very active research field involving scientists from various disciplines, especially physicists, mathematicians, engineers and computer scientists.
It has a long history which goes back to L. Euler who introduced the celebrated Euler equations (both the compressible and the incompressible ones) in 1755. It is worth recalling that L. Euler mentioned immediately the analytical (we would say mathematical nowadays) difficulties associated with these models. And, two centuries and a half later, some of these difficulties are still important issues which generate a lot of activity in Mathematics and Scientific Computing: let us mention, in particular, the questions of the formation of singularities in three-dimensional solutions of the incompressible Euler equations, of the mathematical structure of solutions of compressible Euler questions, etc.
 Despite fundamental contributions by outstanding scientists and mathematicians like Riemann, von Karman, Leray, von Neumann, Hoff, Lax, etc., basic issues concerning the mathematical understanding of Fluid Mechanics equations is still far from being satisfactory.
 In addition, computers and the possibility of numerical simulations since the late 40's and von Neumann's work have changed drastically the field where the theoretical mathematical analysis is now intimately connected with numerical experiments and simulations. Furthermore, the progress in power and speed of modern computers has led engineers and scientists into considering more and more complex fluid flows and mathematical models.
 Therefore, the complexity of realistic fluid flows together with the need of advanced tools for their numerical simulation demand for a systematic investigation of the fundamental mathematical models.
 The latter include not only the classical nonlinear partial differential equations, such as Euler or Navier-Stokes equations, but also their modifications which are motivated by the need of reducing their numerical complexity. Let us mention, for instance, the various turbulence models, or coupled models such as those involving the interaction between viscous and inviscid flows, fluids and structures, etc.
 Our goal in this programme is to improve the mathematical understanding of these models as well as  of the various existing numerical methods and approaches. In addition we expect to propose new methods based upon the already known fundamental mathematical properties of the Fluid Mechanics equations.
 Research on such challenging mathematical and computational issues would greatly benefit from a concerted action among several European groups.
 We describe below, in particular, the objectives of the proposed programme, the problems and methods that will be investigated, as well as the activities that we are going to undertake. Before we do so, we wish to make a few more specific remarks on the subject which better motivate this proposal and outline why this is a good time to promote this concerted action.

1.2 Remarks on some recent mathematical progress on fluid flows

 We shall  not attempt to review all recent progress on fluid flows but we shall only give two examples indicating  some possible research directions for the groups concerned with this proposal. The first example concerns compressible Navier-Stokes equations. In the past ten years, much progress has been made on the one-dimensional case by Kezhilzhov and his collaborators, D. Hoff and D. Serre, and, very recently, global weak solutions have been obtained by P.L. Lions. These mathematical achievements bring the subject to a situation somewhat comparable to the one for incompressible flows  after J. Leray's work. In view of the impact of J. Leray's solutions and, more importantly, proofs and methods from the numerical analysis of incompressible Navier-Stokes equations, we may expect considerable progress on the mathematical understanding of numerical methods for compressible Navier-Stokes equations.
The second example we want to mention concerns gas-dynamics equations and compressible Euler equations, which represent one of the fundamental examples of hyperbolic conservation laws. This is obviously a subject  where the interplay between mathematical theory, the construction of numerical methods, and computer experiments, has played a crucial role in the development of the field. The mathematical theory of compensated compactness of L. Tartar (and F. Murat, R. Di Perna) combined with the earlier idea of "Lax entropies" has had and is still having many applications to the numerical analysis of finite differences and finite volumes methods. More recently, the kinetic formulation of such equations and the related idea of the so-called Boltzmann schemes are beginning to receive much attention leading to truly multidimensional schemes.
 Let us emphasise the fact that the above two examples illustrate some of our goals in this proposal and the strong connections between progress in the mathematical understanding, the understanding of existing numerical methods and the design of new numerical approaches. In the next subsection, we present more examples of efficient numerical approaches that require a better mathematical understanding.

1.3 Remarks on some recent and perspective approaches for the  numerical simulation of fluid flows

Numerical methods have found widespread application in the simulation of a large variety of flows of both scientific and industrial interest. Engineers and physicists have brought remarkable ideas to the subject, and adopted advanced mathematical tools for its investigation.
We will describe two families of problems which, although not exhaustive of the field of our interest, are nonetheless of paramount importance and useful to illustrate the kind of contribution that this project could bring to the subject.

1.3.1 Modeling and simulation of turbulent and reactive flows

 Direct numerical simulation (DNS) is a method in which all scales of motion of a turbulent flow are computed. A DNS must include everything from the large integral scale to the dissipative (viscous, or Kolmogoroff) scale.
It is well known that, apart from flows at moderate Reynolds number, turbulent behaviour can hardly be reproduced by the DNS. Indeed, for flows with high Reynolds numbers, the Kolmogoroff turbulent scales are much smaller than the affordable time-step and mesh- spacings, which are the time and length scales of the discretized problem. Thus the turbulent diffusion cannot be captured by DNS. The level of the physical model needs therefore to be reduced  through the introduction of suitable turbulence models. Instances are provided by the Baldwin-Lomax algebraic method, the k-e method (Launder, Schumann, Spalding, Pironneau,...) for the conservation of kinetic energy and its dissipation rate, the Reynolds stress model, the renormalization group theory (Orszag, Yakot,...), and  the large eddy simulation (LES) method (Ferziger, Rogallo, Germano, ..). LES provides a direct numerical simulation of large vortical structures but models the exchange mechanism that transfers energy from high to low frequencies. Thus, when LES is adopted, the largest scales of motion are represented explicitly while the small scales are treated by some approximate parameterization or model. Large eddy simulations are three dimensional and time-dependent, and therefore costly, although they are often much less costly than DNS of the same flow.
In turbulence modeling and in LES, we are facing a difficult problem with the correct treatment of the boundary conditions. Experiments reveal that coherent structures exist there and herefore, a major modeling effort has to be carried out in order to represent the corresponding physical phenomena.
More generally, the use of  advanced multiscale numerical methods which can take care of flows with multiple scales, is of great help not only for their simulation  but also for the investigation of  the computed flow-fields (e.g., by wavelet transformation), and, consequently, provide a tool for carrying out a-posteriori error analysis and adaptivity.
 Chemical reactions or combustion  (e.g. premixed turbulent combustion or diffusion controlled combustion) add further complexity to the flow structure. Montecarlo techniques for the transport of the probability density function (pdf) are usually adopted for the simulation of turbulent combustion, in alternative to the DNS.
 One of the aims of our project is bringing together mathematicians, physicists and engineers to cooperate on the following items: modeling the physics of turbulent and reactive flows, develop a mathematical analysis of the associated partial differential equations, simulate them  by numerical methods, and investigate their application to real life problems.


1.3.2 Shock-capturing methods for nonlinear conservation laws

 As pointed out by P. Lax in a recent historical retrospective, interest in shock waves in the United States started during the Second World War, culminating in the well-known, influential treatise on supersonic flow and shock waves by Courant and Friedrichs. A systematic effort to compute flows with shock waves was initiated during the war by von Neumann at Los Alamos. Starting with an idea conceived in 1944, von Neumann and Richtmyer in 1950 invented a numerical scheme that, with  the aid of artificial viscosity, captured shocks as rapid internal transitions.
Upon Richtmyer's influential monograph on numerical methods for solving initial value problems, Lax's contribution laid in the convergence and accuracy of difference approximations of solutions of hyperbolic equations. For linear equations the main results are that stability implies convergence, and that stability follows from the stability of the "frozen" difference schemes with constant coefficients, something that can be checked by Fourier transformation and matrix analysis. For nonlinear equations proof of convergence is much harder, and for realistic, complicated problems analysis furnishes "design principles". One of the important design principles is that conservation laws should be approximated by difference equations in conservation form, employing a numerical flux that approximates the physical flux. It was proven that if approximate solutions constructed by such a scheme converge strongly, say in the L1 sense, then the limit satisfies the original conservation laws.
Another useful design principle is that if an entropy function can be defined for a given system of conservation laws, then the numerical method employed to construct approximate solutions should avoid increasing entropy.
Nowadays, after McCormack, Baldwin, Harten and Jameson in the 70's, almost all the efficient numerical schemes are nonlinear, i.e. they introduce nonlinear dissipation. Numerical dissipation provides stability, damps unwanted oscillations generated by second and higher order schemes around discontinuities and stiff layers. It is introduced either by centered approximations (using second and fourth order smoothers, or even spectral viscosity), or resorting to uncentered, upwind like schemes.
Upwind techniques introduce physical information in the numerical discretization (the direction of  propagation of waves) (Courant, Isaacson and Rees, Steger and Warming, Van Leer,..). At a first stage of their evolution they have been designed so to be conservative, and later on to take into account information on exact solutions as well, by generalizing Godunov's idea of  solving one-dimensional Riemann's problems at interfaces (also numerically, after Roe and Osher).
The total variation diminishing schemes (TVD), and the essentially non oscillatory schemes (ENO), have then been introduced (Harten , Engquist, Osher, ...) with the purpose of  getting high resolution schemes without loosing the control on the total variation of the numerical solution (smoothing out the potential oscillations), and guaranteeing high order away from the shock (including the extrema of the flow flux). The extension of these techniques to multidimensional vector problems is still an active area of investigation, especially when using finite volume or finite element (unstructured) grids, where multidimensional, cartesian splitting is no longer straightforward (Baines, Hughes, Deconinck, Dervieux, ...). In our days,  there is a cross fertilization of ideas between the finite element approach using nonlinear stabilization a' la SUPG (Johnson, Szepessy, Tezduyar, Deconinck, Roe,..) or the so-called discontinuous finite element method (Raviart, Lesaint, Cockburn, Shu, ...) and the finite volume method using node centered unknowns and dual cells. In particular, this allows the use of the rich finite element functional framework (including the compensated compactness technique, or the use of equivalent principles between SUPG methods and Galerkin methods with bubble corrections recently introduced by Brezzi, Franca, Hughes,...) to analyze shock capturing schemes on unstructured grids.