An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction

Hybrid Discrete Ordinates Solver for the Radiative Transport Equation using Second Order Finite Volume and Discontinuous Galerkin

“…The next step is to compute an approximation f ∈ V u of ψ n+1 from f u and f c . The strategy in [12,32] is to let f = f u + Rf c , where R : V c → V u is a "relabeling operator". Here, we instead follow [11] and solve the following approximation of (3.1.6):…”

“…It was observed in [11,12] that hybridization in angle can achieve more accurate numerical solutions than a standard approach by simply increasing the angular resolution in the uncollided equation while reducing the resolution in the collided equation. The results in Figure 4.2 are consistent with this observation, and for a standard DG discretization, the results of such a strategy are shown in the middle two columns of Table 4.4.…”

“…In this context we show that, under reasonable assumptions, the spatial hybrid captures numerically the diffusion limit, which is realized asymptotically in the limit of infinite scattering. We show how the spatial hybrid can be used in conjunction with existing angular hybrid methods, by combining it with the discrete ordinate hybrid from [12]. Finally, we demonstrate that the spatial hybrid is more efficient, in terms of memory usage and time to solution, than the standard DG discretization of the transport equation.…”

Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere

“…In this work, we consider semi-implicit, iterative, multi-level temporal integration methods based on Spectral Deferred Corrections (SDC) that are easily extended to high-order and also serve as a first step toward constructing parallel-in-time integration methods for the atmospheric dynamics based on PFASST.SDC methods are first presented in Dutt et al (2000) and consist in applying a sequence of low-order corrections -referred to as sweeps -to a provisional solution in order to achieve high-order accuracy. Single-level SDC schemes have been applied to a wide range of problems, including reacting flow simulation (Bourlioux et al, 2003;Layton and Minion, 2004), atmospheric modeling (Jia et al, 2013), particle motion in magnetic fields (Winkel et al, 2015), and radiative transport modeling (Crockatt et al, 2017). In Jia et al (2013), a fully implicit SDC scheme is combined with the Spectral Element Method (SEM) to solve the shallow-water equations on the rotating sphere.…”

Scientific data interpolation with low dimensional manifold model

“…The simulated quantity is a distribution function that depends on five 195 independent variables: two spatial, two angular, plus time. The data used here was generated using the algorithm described in [43] which combines a third-order space-time discretization (discontinuous Galerkin in space and integral deferred correction in time) and an angular discretization based on a tensor product collocation scheme. 200 We consider for this problem two quantities of interest.…”