Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

“…This result can be extended to essentially all AP schemes, although the specific proof is problem dependent. We refer to AP schemes for kinetic equations in the fluid dynamic or diffusive regimes [2,7,14,32,[40][41][42]44,45,[47][48][49]. The AP framework has also been extended in [15,16] for the study of the quasi-neutral limit of Euler-Poisson and Vlasov-Poisson systems, and in [19,21,34] for all-speed (Mach number) fluid equations bridging the passage from compressible flows to the incompressible flows.…”

“…In [42], the second lf term in absorbed into the time derivative, which removes the stiffness, and then Q(f) is approximated by the Wild Sum which is truncated at finite terms with the remaining infinite series replaced by the local Maxwellian in order to become AP. If one is just interested in removing the stiffness, one can just approximate the right hand side of (2.4) by…”

“…(1.1) with the Knudsen number e > 0 depending on the space variable in a wide range of mixing scales. This kind of problem was already studied by several authors for the BGK model [20] or the radiative transfer equation [42]. In this problem, e : R#R þ is given by represented with crosses (x) and the numerical solution to the compressible Navier-Stokes system (lines): evolution of (1) the density q, (2) Fig.…”

Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes

“…In addition to the difficulty of numerical stiffness arising due to the small scaling, improper underresolved numerical solution often fail to capture the hydrodynamic drift-diffusion limit. Earlier study on numerical methods for transport or kinetic equations indicates that, in order for the underresolved numerical approximation to capture the correct diffusive behavior, the scheme should be asymptotic preserving (AP), in the sense that the asymptotic limit that leads from the transport or kinetic equations to the diffusion equations should be preserved at the discrete level [Ada,Jin,JL1,JL2,JPT1,JPT2,Kl1,Kl2,LMM,LM,Mil,NP1,NP2].…”

“…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 mean free path. Such a numerical technique was extended to general transport equations with isotropic collision kernels, by using parity formulation of the transport equation [JPT2].…”