In this short course we will focus on the new classes of numerical multi-scale methods, which couple models for different scales in the same computation. Continuum simulations of solids or fluids for which some atomistic information is needed are typical example of such problems with very large ranges of scales. We will base the study of these new classes of numerical methods on the heterogeneous multi-scale method, which is a framework for analysis and design of multi-scale algorithms. In these techniques local micro-scale simulations on small domains supply missing data to a macro-scale simulation on the full domain. Examples from solid and fluid mechanics and from stiff dynamical systems will be given. As a background for this material we will review relevant elements of numerical complexity theory, analytic homogenization and model reduction and of traditional numerical multi-scale methods as, for example multi-grid and the fast multi-pole method.

This minicourse will be devoted to the modelization of water-waves, with a special focus on coastal flows. After exhibiting the physically relevant parameters which determine the nature of the flow, we will derive various asymptotic models from the water-waves equations. The nature of these models depends of course heavily on the respective size of the above mentionned parameters, but one can also obtain various mathematically 'equivalent' models for a given physical setting. We will show that other criteria can however be used to show that these asymptotically equivalent models are not physically of equivalent interest.

I will present an overview of fluid flow modelling at three scales: a large scale at which Navier-Stokes, Stokes, and Euler equations are relevant, a smaller scale at which models such as the lubrication model are relevant, and the microscale. This will include a discussion of passing initial data, boundary conditions, and geometric effects along the heirarchy. Lecture notes will be provided.

The most successful and most frequently used schemes for solving conservation laws are the Finite Volume and the Discontinuous Galerkin schemes. The Finite Volume schemes are special cases of the Discontinuous Galerkin ones. But there are important differences, in particular for solving the MHD equations. A main problem for the MHD equations concerns the divergence free magnetic field. Usually the initial data for the magnetic field are divergence free and the exact solution will keep this property for all times. But due to numerical errors many numerical magnetic fields will lose this property for later times. For the discretization of the divergence-free property of the magnetic field many attempts can be found in the literature. In this lecture we will discuss a heuristic method and a rigorous error estimate, based on a Discontinuous Galerkin method, for solving this problem.

Part I of this introduction explains the oscillation phenomenon from nonconvex minimization in 1D examples named after Bolza, Young, and Tartar. In the finite element discretisation, the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures which are hardly computable.

Some macroscopic or effective quantities, however, are well-posed and the target of an efficient numerical treatment as outlined in part II of this introduction. The relaxation finite element method is introduced for macroscopic simulations. Moreover, adaptive mesh-refining algorithms for the finite element method for the effective equations () are discussed. For some class of convexified model problems, a priori and a posteriori error control is available with an reliability-efficiency gap. Nevertheless, convergence of some adaptive finite element schemes is guaranteed. Applications involve model situations for (1), (2), and (3) where the relaxation is provided by a simple convexification.

I will first talk about the "Riemann Problem" for a junction,with the class of "second order " model introduced several years ago in Aw-Rascle, SIAP, 2000 ("Resurrection"), in connection with the celebrated ("Requiem ...") paper of Daganzo in 95.

I will show the differences with first order models or e.g. the solution proposed by Garavello-Piccoli and I will emphasize the homogenization whuch arises na- turally in this problem, especially with maximal flux criteria. If time allows, I will then introduce a new (Lagrangian) Hybrid model recently introduced with S. Moutari, which among other things presents no difficulty to conserve the mass, and still always provides a microscopic description of a fixed Eulerian region.

The basic equation for the homogenization of linear hyperbolic models is the simple transport equation

Production flows involving a large number of items and a large number of production stages are a typical multiscale modeling problem. At the most elementary level they are modeled as stochastic discrete event simulations, at a more aggregate level they are called fluid models (ODEs!), at a yet higher level they can be modelled as PDEs (moment equations) - or alternatively as Equation Free Models. We will present several PDE models based on mass conservation, diffusion and capacity limited fluxes. Comparisons between discrete event simulations, factory data, numerical solutions of the PDEs and queuing network models are made. Results of an Equations Free approach are compared. First attempts and open questions relating to boundary control of these equations are also discussed.

This talk will deal with the modeling of a plasma in the quasi-neutral limit using the two-fluid Euler-Poisson system. In the quasi-neutral limit, explicit discretizations of this model suffer from severe numerical constraints related to the small Debye length and large plasma frequency. The satisfaction of these constraints requires huge computational resources which make the use of explicit methods almost impracticable. I will present an implicit scheme such that, in the quasineutral limit, a discretization of the quasineutral Euler model is recovered. Such a property is referred to as ``asymptotic preservation'': the scheme preserves the asymptotic limit.
Additionally, in spite of being implicit, the scheme has the same computational cost as the standard explicit strategy, the resolution of the Poisson equation being replaced by that of a different (but equivalent) elliptic equation, which is not more difficult to solve.