Asymptotic preserving numerical schemes for a non‐classical radiation transport model for atmospheric clouds

Abstract

We present numerical schemes for the P1-moment and M1-moment approximations of a non-classical transport equation modeling radiative transfer in atmospheric clouds. In contrast to classical radiative transfer, the photon path-length is introduced as an additional variable and serves as pseudo-time in this model. Because clouds may have optically thick regions, we introduce a diffusive scaling and show that the diffusion limits of the moment models and the original equations agree. Furthermore, we show that the numerical schemes also preserve the diffusion asymptotics as well as the set of admissible and realizable states, both for the explicit and the implicit discretization of the pseudo-time variable. A source iteration-like method is proposed, and we observe that it converges slowly in the optical thick case, but a suitable initialization can help to overcome this problem. We validate our method in 1D and present simulation results in the 2D-case for real cloud data.In case of a collision event, 0 < c < 1 denotes the probability that the photon is scattered. Then, a photon with direction v changes its direction to v 0 with a probability given by the scattering kernel .v, v 0 /. With a probability of 1 c, the particle is absorbed. After scattering, the particle travels a certain distance until the next collision event occurs.A probability density p for the distance between two collisions can be defined in terms of the scattering frequency † t and vice versaSolving the full linear Boltzmann equation is numerically costly because of its high dimensionality. Hence, we derive moment models for Equation (1) in order to reduce complexity. Introducing the first three angular moments of function given by

A new deviational asymptotic preserving Monte Carlo method for the homogeneous Boltzmann equation

“…It is then very important to find numerical methods which avoid the bottleneck caused by a growing number of collisions or equivalently by the stiffness of the equations [29]. A well-known class of schemes which is designed expressly for treating multiscale problems is the Asymptotic Preserving (AP) one [3,20,28,37,43,44,27,30,26,56,33,38,40,41]. Their main interest is that time steps are not constrained to the fast scale dynamics and that they automatically degenerate to consistent discretizations of the limiting models when the parameters which characterize the microscopic behaviors go to zero.…”

Asymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling

“…This strategy is somehow optimal in terms of efficiency since the expensive kinetic model is used only when necessary, but it requires to connect the macro and micro models which is not an easy task at both mathematical and numerical levels, even if efficient approaches have been recently developed [27,56]. Another way to handle such multiscale nature of the phenomena is to use asymptotic preserving (AP) methods [7,8,17,26,40,31,53,34,39,41,42,43,44,50] which enable to overcome the numerical stiffness and to use time and space steps which are independent of the stiffness parameters which characterize the fast scales. However, while this approach permits to solve the problem related to the choice of the small time and space steps, it does not overcome the cost related to the solution of the kinetic model even in the equilibrium regions where a much less complex asymptotic model could be used.…”

A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

“…When the different scales are less clearly delimited, we have to develop kinetic schemes that naturally reduce to good approximations of the macroscopic problem when the system goes near its equilibrium, and overcome the stiffness. Such schemes are often called Asymptotic-Preserving (AP), see [14,17,16,12,8,18,19,20,21,15,24,4]. Mainly, the numerical cost remains comparable to the one of the non-stiff kinetic problem, even when ε ≪ 1.…”