Abstract: In the Eulerian description of kinetic/fluid dynamics such as kinetic equation and compressible gas dynamics, numerical dissipations are necessarily introduced to have a stable discretization. One severe difficulty in multi-scale kinetic/fluid simulations is the amplification and accumulation of these numerical dissipations due to the dynamics in small scale motions such as fast waves and small mean free path. Another known difficulty in multi-scale kinetic/fluid simulations is the inhibitively small time stepping. In this talk, we will show that with some proper decompositions of the underlying, governing equations, these two difficulties can be removed in the neutron transport equation uniformly with respect to the mean free path, and in the gas dynamics uniformly with respect to the Mach number. In connection with the neutron transport equation, I will also present some PDE results on estimation of first return time/short path statistics of Pearson random walk under isotropic uniform incidence form the boundary. |