In processes involving multi-phase flow in highly heterogeneous media, such as oil recovery by gas injection, mobile phases will seek high-permeability flow paths. Therefore, it is essential that models for such processes effectively account for these paths. For this purpose, we have developed a computational framework for flow solvers based on adapted Cartesian grids that are equipped with multi-point flux approximations obtained with specialized transmissibility upscaling methods.

For gridding, we propose using Cartesian Cell-based Anisotropically Refined (CCAR) grids, which inherit the ease of Cartesian grids while providing rapid transition between coarse and fine scales to resolve fine-scale features accurately and efficiently. We present an iterative algorithm for automatically generating such grids based on geological data and information from global coarse-scale flow simulations.

For upscaling, we discuss a local transmissibility upscaling method, called Variable Compact Multi-Point (VCMP), that uses spatially varying and compact multi-point flux stencils. The stencil weights are chosen so as to reproduce generic local flow problems accurately, while remaining as close as possible to a two-point flux for the sake of robustness. The inherent flexibility of VCMP can also be exploited to ensure that the solution of the resulting system satisfies a discrete maximum principle.

We conclude with application of these gridding, upscaling and discretization methods, originally designed for single-phase flow, to two-phase flow, which requires enhancing our adaptive mesh refinement scheme in order to accurately resolve rapidly advanacing saturation fronts. We show that adaptivity allows such accurate resolution by upscaling only single-phase parameters, thus avoiding the significant computational expense of multi-phase upscaling.

In this lecture, we present a method which has broad applicability to studies of nonlinear stability of periodic traveling-wave solutions for equations of KdV-type. In particular we obtain the existence and stability of a family of periodic traveling-wave solutions for the Benjamin-Ono equation via the classical Poisson summation theorem and positivity properties of the Fourier transform.

This is a report on a joint work with Isabelle Catto, Norbert Mauser and Saber Trabelsi. The Multiconfiguration time dependent Hartree Fock Method (MCTDHF) is a nonlinear approximation of a linear system of /N/ quantum particles with binary interaction. It combines the principle of the Hartree Fock and the Galerkin approximation. The main difficulty is the introduction of a global (in space) density matrix $\Gamma(t) $ which may degenerate. By construction this approximation formally preserves the mass and the energy of the system. The conservation of energy can be used to balance the singularities Coulomb potential and to provide sufficient conditions for the global in time invertibility of $\Gamma(t)$.

In numerical computations this matrix is very often regularized (changed into $\Gamma(t) +\epsilon(t)$). In this situation the energy is no more conserved
and the mathematical analysis done in $L^2$ relies on Strichartz type estimates.

Since the seminal paper by Perona and Malik nonlinear diffusions have successfully been used for various image processing tasks. They also have attracted steadfast interest in the mathematical community. In this talk we will give an historical overview of the developments on the Perona-Malik equation and describe two new nonlinear diffusions which resolve the main mathematical shortcomings of Perona-Malik without sacrificing but rather enhancing the cherished practical qualities of the Perona-Malik model. The new equations, while
well-posed and purely diffusive, exhibit a non trivial dynamical behavior, which makes them mathematically interesting and practically
effective.

The question of global regularity for the 3D Navier-Stokes equations is a major open problem in applied analysis. It is well-known that the existence and uniqueness of strong solutions could be obtain under suitable additional assumptions. In this talk I will review some old and new results about the sufficient conditions for the global regularity to the 3D Navier-Stokes equations.