Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II

Abstract

In a recent article (Larsen, Morel, and Miller, J .Comput. Phys. 69, 283 (1987)), a theoretical method is described for assessing the accuracy of transport differencing schemes in highly scattering media with optically thick spatial meshes. In the present article, this method is extended to enable one to determine the accuracy of such schemes in the presence of numerically unresolved boundary layers. Numerical results are presented that demonstrate the validity and accuracy of our analysis.

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.…”

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].…”

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

“…The approximation of the low-order equations is based on the linear discontinuous (LD) method that was formulated for the transport equation [18,19,15]. We introduce the spatial mesh…”