Parallel kinetic schemes for conservation laws, with large time steps

TL;DR

Addresses the CFL-limited nature of explicit DG methods for hyperbolic conservation laws and delivers an unconditionally stable, CFL-free kinetic DG framework with explicit-like complexity. Based on a vector BGK representation with $W=\sum_{k=0}^d F_k$ and Maxwellians $M_k(W)$, it introduces a subcharacteristic condition and a transport-relaxation scheme that reduces to independent transport equations coupled by relaxation. A second-order accurate scheme is obtained via composition of transport operators ($\mathcal{M}\circ\mathcal{M}$), and a subdomain iterative strategy enables scalable parallelism on distributed memory machines; the source terms are handled with Crank–Nicolson and an implicit relaxation. Numerical experiments on Maxwell's equations validate stability at large time steps, planewave and wire tests, and a large-scale antenna–mannequin simulation, with competitive computation times against FDTD/CLAC solvers.

Abstract

We propose a new parallel Discontinuous Galerkin method for the approximation of hyperbolic systems of conservation laws. The method remains stable with large time steps, while keeping the complexity of an explicit scheme: it does not require the assembly and resolution of large linear systems for the time iterations. The approach is based on a kinetic representation of the system of conservation laws previously investigated by the authors. In this paper, the approach is extended with a subdomain strategy that improves the parallel scaling of the method on computers with distributed memory.

Figures (18)

• Figure 3.1: Notation for the Discontinuous Galerkin approximation.
• Figure 3.2: Example of a mesh $\mathcal{M}$ (left panel) and its associated graph $G$ (right panel). The nodes of the graph correspond to the cells of the mesh. Two additional, fictitious nodes are considered: the upwind node (in orange) and the downwind node (in blue). The solution can be explicitly computed by following a topological ordering of a Direct Acyclic Graph (DAG) using Breadth-First Search, e.g. 3, 7, 0, 15, 1, etc. In addition, the parallel capabilities of the method are visible on the graph: first, cells $3$ and $7$ can be computed in parallel; then cells $0$, $15$ and $1$ can be computed in parallel, etc.
• Figure 5.1: Subdomain algorithm, when the subdomain decomposition is aligned with the transport velocity. In this case, the iterative algorithm reaches the exact solution in at most two iterations. During the first iteration, in the left panel, the boundary values of the subdomains are updated. During the second iteration, in the right panel, the correct boundary values are transported. The purple color corresponds to the transport of a correct value, while the red color corresponds to the transport of a wrong value.
• Figure 5.2: Subdomain algorithm, in a generic subdomain decomposition, with corners shared by several subdomains. In this case, the iterative algorithm reaches the exact solution in at most three iterations. First iteration, left panel: the boundary value on $\partial\Omega_{2}^{-}$ is updated. Second iteration, center panel: the boundary value on $\partial\Omega_{3}^{-}$ is updated. Third iteration, right panel: the correct boundary value is transported. The purple color corresponds to the transport of a correct value, while the red color corresponds to the transport of a wrong value.
• Figure 5.3: Stability of the subdomain iterative algorithm applied on a mesh of the unit cube with 8 subdomains. Left: structure of the subdomains; Top right: scheme with two iterations; Bottom right: scheme with three iterations. We observe that the iterative algorithm is stable, even with large time steps, but that three iterations seem to be necessary.

Theorems & Definitions (3)

• Remark
• Proposition 1
• proof