Luc Mieussens 教授 特別講演会

In this talk, I will present two new methods to describe systems of particles that contain both microscopic (kinetic) and macroscopic scales.

The first one provides a smooth transition between a kinetic and a hydrodynamic domain. The idea is to use a buffer zone, in which both hydrodynamics and kinetic equations will be solved. The solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations. We use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone, thus no boundary condition is needed at the transition point. Consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations.

The second method presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the non-equilibrium part of the distribution function. This equation is solved only locally and is related to the fluid equation through a downscaling effect. The method does not need to find an interface condition. We show our approach applies to problems that have a hydrodynamic time scale as well as to problems with diffusion time scale.

Simple numerical schemes are proposed to discretized our models, and several numerical examples are used to validate the methods.

These works have been obtained in collaboration with Pierre Degond (Toulouse, France), Giacomo Di Marco (Ferrara, Italy), Shi Jin (Wisconsin, USA), and Jian-Guo Liu (Maryland, USA).