Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations

  • Authors:

  • Shi Jin

  • Affiliations:

  • -

  • Venue:
  • SIAM Journal on Scientific Computing
  • Year:
  • 1999

Quantified Score

Hi-index 0.07

Visualization

Abstract

Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter, from the rarefied kinetic regimes to the hydrodynamic diffusive regimes. An earlier approach in [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] reformulates such systems into the common hyperbolic relaxation system by Jin and Xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time-splitting method to solve the relaxation system. Here we observe that the combination of the two time-split steps may yield hyperbolic-parabolic systems that are more advantageous, in both stability and efficiency, for numerical computations. We show that such an approach yields a class of asymptotic-preserving (AP) schemes which are suitable for the computation of multiscale kinetic problems. We use the Goldstein--Taylor and Carleman models to illustrate this approach.