Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439 we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.