Efficient Spacetime Meshing with Nonlocal Cone Constraints

TL;DR

This paper introduces an efficient advancing front algorithm for spacetime meshing in DG methods that adapts to local wavespeed variations and nonlocal cone constraints, enabling constant-time solution per element.

Contribution

The paper presents a novel algorithm that guarantees causal, well-structured spacetime meshes for nonlinear hyperbolic PDEs with discontinuous wavespeeds, improving upon previous methods.

Findings

Algorithm guarantees causal patches with bounded size.

Mesh adapts to local wavespeed and spatial gradation.

Enables constant-time solution per element in spacetime mesh.

Abstract

Spacetime Discontinuous Galerkin (DG) methods are used to solve hyperbolic PDEs describing wavelike physical phenomena. When the PDEs are nonlinear, the speed of propagation of the phenomena, called the wavespeed, at any point in the spacetime domain is computed as part of the solution. We give an advancing front algorithm to construct a simplicial mesh of the spacetime domain suitable for DG solutions. Given a simplicial mesh of a bounded linear or planar space domain M, we incrementally construct a mesh of the spacetime domain M x [0,infinity) such that the solution can be computed in constant time per element. We add a patch of spacetime elements to the mesh at every step. The boundary of every patch is causal which means that the elements in the patch can be solved immediately and that the patches in the mesh are partially ordered by dependence. The elements in a single patch are coupled because they share implicit faces; however, the number of elements in each patch is bounded. The main contribution of this paper is sufficient constraints on the progress in time made by the algorithm at each step which guarantee that a new patch with causal boundary can be added to the mesh at every step even when the wavespeed is increasing discontinuously. Our algorithm adapts to the local gradation of the space mesh as well as the wavespeed that most constrains progress at each step. Previous algorithms have been restricted at each step by the maximum wavespeed throughout the entire spacetime domain.