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Abstract: Spacetime Discontinuous Galerkin DG methods are used to solve hyperbolicPDEs describing wavelike physical phenomena. When the PDEs are nonlinear, thespeed of propagation of the phenomena, called the wavespeed, at any point inthe spacetime domain is computed as part of the solution. We give an advancingfront algorithm to construct a simplicial mesh of the spacetime domain suitablefor DG solutions. Given a simplicial mesh of a bounded linear or planar spacedomain M, we incrementally construct a mesh of the spacetime domain M x0,infinity such that the solution can be computed in constant time perelement. We add a patch of spacetime elements to the mesh at every step. Theboundary of every patch is causal which means that the elements in the patchcan be solved immediately and that the patches in the mesh are partiallyordered by dependence. The elements in a single patch are coupled because theyshare implicit faces; however, the number of elements in each patch is bounded.The main contribution of this paper is sufficient constraints on the progressin time made by the algorithm at each step which guarantee that a new patchwith causal boundary can be added to the mesh at every step even when thewavespeed is increasing discontinuously. Our algorithm adapts to the localgradation of the space mesh as well as the wavespeed that most constrainsprogress at each step. Previous algorithms have been restricted at each step bythe maximum wavespeed throughout the entire spacetime domain.



Author: Shripad Thite

Source: https://arxiv.org/







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