A high-order / low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Abstract

Dynamical low-rank (DLR) approximation methods have previously been developed for time-dependent radiation transport problems. One crucial drawback of DLR is that it does not conserve important quantities of the calculation, which limits the applicability of the method. Here we address this conservation issue by solving a low-order equation with closure terms computed via a high-order solution calculated with DLR. We observe that the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. We also apply the linear discontinuous Galerkin method for the spatial discretization to obtain the asymptotic limit.We then demonstrate with the numerical results that this so-called high-order / low-order (HOLO) algorithm is conservative without sacrificing computational efficiency and accuracy.

Multi-Frequency Implicit Semi-analog Monte-Carlo (ISMC) Radiative Transfer Solver in Two-Dimensions (without Teleportation)

“…First, we repeat here the low temperature, constant opacity hohlraum problem presented in Brunner [2], (and with a slight modifications later, in [12,23]). This problem was the first benchmark that was offered of its hohlraum type.…”

A rank-adaptive robust integrator for dynamical low-rank approximation

“…The reference solution to this problem is given by the standard de facto Ganapol's benchmark test [10] and this problem has been investigated for dynamical low-rank approximations in [25,24]. As time increases, the scalar flux Φ(t, x) = 1 −1 f (t, x, µ) dµ moves to the left and right side of the spatial domain, showing a discontinuous (or shock) profile at the front.…”

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

“…Common methods require high computational costs or parameter tuning to yield a satisfactory approximation. Uses of dynamical low-rank approximation for this benchmark are [37,36,8], where it is observed that high ranks are needed to achieve a desired level of accuracy. Nevertheless, in comparison to classical methods, DLRA yields reduced runtimes and memory requirements.…”

“…More exact dose calculation can be achieved by an appropriate Monte Carlo (MC) algorithm, where individual interacting particles are directly simulated [4]. However, while recent performance-tuned MC The efficiency of dynamical low-rank approximation has been demonstrated in several applications, including kinetic theory [14,15,37,36,16,12,13,8,30,11]. Two main challenges of DLRA in the context of kinetic theory and radiation transport specifically are the preservation of mass as well as capturing the asymptotic limit.…”

“…Two main challenges of DLRA in the context of kinetic theory and radiation transport specifically are the preservation of mass as well as capturing the asymptotic limit. Methods to guarantee mass conservation include re-scaling strategies [37], a high-order low-order (HOLO) decomposition [36] and the incorporation of certain basis functions in the tangent space of low-rank functions [13]. An asymptotic preserving scheme for dynamical low-rank approximation has been proposed in [11].…”