Domain decomposition dynamical low-rank for multi-dimensional radiative transfer equations

Abstract

In this paper, we propose a domain decomposition dynamical low-rank method to solve high-dimensional radiative transfer problems and similar kinetic equations. The algorithm uses a separate low-rank approximation on each spatial subdomain, which means that, for a given accuracy, we can often use a smaller overall rank compared to classic dynamical low-rank methods. In particular, we can solve problems with point sources efficiently, that for classic algorithms require almost full rank. Our algorithm only transfers boundary data between subdomains and is thus very attractive for distributed memory parallelization, where classic dynamical low-rank algorithms suffer from global data dependency. We demonstrate the efficiency of our algorithm by a number of challenging test examples that have both very optical thin and thick regions.

Figures (15)

• Figure 1: Illustration of block domain decomposition.
• Figure 2: Inflow and outflow for subdomain $\Omega_C$ during the x advection step.
• Figure 3: Setup of the lattice test problem.
• Figure 4: We show $\rho (t,x,y)$ obtained by solving the lattice test problem with a classic dynamical low-rank algorithm (i.e. with a single low-rank representation for the entire domain) on the left and with the domain decomposition low-rank algorithm \ref{['alg::ddlr_algorithm']} on the right. For the latter, $7 \times 7$ subdomains, each containing $36$ grid points in both the $x$ and $y$ directions, have been used.
• Figure 5: We show the time evolution of the rank for the classic low-rank simulation of the lattice test problem. As an adaptive rank scheme we used the scheme described in Hochbruck_rankadaptivity with $\text{tol}=3\cdot10^{-5}$.