High-Order Upwind Methods for Second-Order Wave Equations on Overlapping Grids

Speaker: Jordan Angel, Science & Technology Corporation, NASA Ames Research Center
September 6, 2018

Presentation

Abstract

Novel high-order upwind schemes for the second-order wave equation are presented. First, we develop schemes for the wave equation written as a first-order in time and second-order in space system. These schemes are applied to Maxwell's equations written in second-order form. Interfaces and physical boundaries are described with composite overlapping structured grids and coupled with interpolation and high-order interface conditions. A normal mode analysis from the theory of Gustafsson, Kreiss, and Sundström shows that these schemes are robustly stable against numerical instabilities that are known to arise from overlapping grid interpolation for nondissipative schemes.

A reformulated upwind scheme is presented that considers several optimizations to the first-order in time system. This reformulated scheme discretizes the second-order wave equation directly and is seen to be much more efficient while still retaining the essential stability properties of the first-order in time scheme.

We conclude by considering future work that may extend the application of the upwind method to linear elasticity of solid mechanics and the harmonic coordinate formulation of general relativity.

Biography

Jordan Angel received his Ph.D. in mathematics from Rensselaer Polytechnic Institute in 2018 under the supervision of Bill Henshaw and Jeff Banks. His work on high-order wave solvers is now a part of the composite grid electromagnetics solver included with Overture, a framework for solving partial differential equations on collections of structured grids. His interests are in high-order methods, wave phenomena, and numerical stability. In July 2018, Jordan joined the LAVA group in Computational Aerosciences Branch at NASA Ames Research Center.

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