Asymptotic preserving schemes for kinetic and related systems.

摘要

This dissertation aims at the development of Asymptotic-Preserving (AP) methods for kinetic and related systems, covering three different topics.;The first topic focuses on a class of penalty based AP methods for Boltzmann type equations. These methods originated from the work of Filbet-Jin, and consequently by Dimarco-Pareschi. We generalize their ideas in several aspects: (i) a Fokker-Planck penalization based AP scheme for the Landau equation is designed; (ii) a successive method which inherits all the advantages of both methods is designed; (iii) the extension to the quantum Landau equation is studied. Plenty of numerical tests are carried out to check the performance of the new methods.;The second topic focuses on the development of AP methods for fluid-kinetic coupling system describing particulate flows. The suspended particles are described by Vlasov-Fokker-Planck equation. The surrounding fluid is modeled by the Euler system, or the incompressible Navier-Stokes system with constant/variable spatial density. The two systems are coupled through momentum and energy exchanges. We design numerical schemes which are able to capture the asymptotic behavior, without requiring prohibitive stability conditions. Again numerical experiments are presented, with several interesting applications.;The last topic focuses on the numerical study of the diffusive limit of run & tumble kinetic models for cell motion due to chemotaxis by means of asymptotic preserving schemes. It is well-known that the diffusive limit of these models leads to the classical Patlak-Keller-Segel macroscopic model for chemotaxis. We show that the proposed scheme is able to accurately approximate the solutions before blow-up time for small parameter. The blow-up of solutions is numerically investigated in all these cases.