Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation

Abstract

We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type t ≤ O( x 2 ) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.

The Riemann problem for the shallow water equations with discontinuous topography

“…To find stationary waves, one needs to determine the roots of a nonlinear equation (see the function ϕ in (3.1)) which is convex and, therefore, can be easily computed numerically. The Riemann solver derived in the present paper should be useful in combination with numerical methods for shallow water systems developed in [3,4,11,10,16,6] for which we refer to the lecture notes by Bouchut [5].…”

Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations

“…In this last step, we also note that ∂ t ρ = 0, see (24) and (25), and that, consequently, u is constant. The fully discrete kinetic scheme summarizes in this step as…”

“…We refer to [32] for a discussion on a semiimplicit version of the proposed scheme. This difficulty has motivated the development of implicit methods, as in [24,25].…”

“…We point out that the strategy differs from the one used in [5,7] where the adopted method, based on well-balanced schemes as introduced in [24,25], is implicit (see also [16]). …”

Large Time Step and Asymptotic Preserving Numerical Schemes for the Gas Dynamics Equations with Source Terms

“…Such property is referred to as an asymptotic preserving (AP) property. Since its introduction in the pioneer work of [23,25], the notion of AP numerical schemes has been investigated and implemented in the past years in a wide range of context stemming from hydrodynamics with radiative transfer [22,8,7,2,4], multiscale kinetics [24], diffusive limit of the transport equation [26,29] to problems similar to our model [6,4,11]. Without being exhaustive, we also refer the reader to [9,16,3,1,26].…”