Upwind methods for second-order wave equations on overlapping grids
AuthorAngel, Jordan B.
Other ContributorsBanks, Jeffrey W.; Henshaw, William D.; Schwendeman, Donald W.; Hicken, Jason;
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AbstractA normal mode analysis from the theory of Gustafsson, Kreiss, and Sundstrm (GKS) shows that these schemes are robustly stable against numerical instabilities that are known to arise from overlapping grid interpolation. A reformulated upwind scheme is also presented that considers several optimizations to the ﬁrst-order in time system. This reformulated scheme discretizes the second-order wave equation directly while still retaining the essential stability properties of the ﬁrst-order in time scheme. We give a formally exact diﬀerential-diﬀerence equation which is used to derive high-order predictor-corrector schemes. Numerical experiments show this scheme is typically an order of magnitude faster that the original upwind scheme. The thesis concludes by considering future work that may extend the application of the upwind method to the elastic wave equation of solid mechanics and the harmonic coordinate formulation of general relativity; This thesis is concerned with novel high-order upwind schemes for the second-order wave equation. A ﬁrst-order in time but second-order in space formulation of the wave equation is considered ﬁrst. Following an approach similar to the original idea of Godunov, an exact solution to a local Riemann problem that incorporates upwinding is embedded into a numerical ﬂux function. Using these upwind numerical ﬂux functions we provide an explicit, conservative, and formally exact single-step update formula. We derive high-order accurate approximations from this exact formula which deﬁnes our upwind method. This scheme is applied to Maxwells equations written in second-order form. Several scattering problems are presented consisting of domains with multiple nondispersive materials and high-order convergence is observed. Interfaces and physical boundaries are described with eﬃcient composite overlapping structured grids which are coupled by interpolation and high-order interface conditions.;
DescriptionAugust 2018; School of Science
DepartmentDept. of Mathematical Sciences;
PublisherRensselaer Polytechnic Institute, Troy, NY
RelationshipsRensselaer Theses and Dissertations Online Collection;
AccessCC BY-NC-ND. Users may download and share copies with attribution in accordance with a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. No commercial use or derivatives are permitted without the explicit approval of the author.;
Except where otherwise noted, this item's license is described as CC BY-NC-ND. Users may download and share copies with attribution in accordance with a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. No commercial use or derivatives are permitted without the explicit approval of the author.