Robust High-Order Methods for Wave Equations in the Time Domain

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The goal of this research is to address basic issues in the development of robust and efficient computational methods for simulating waves. Problems governed by wave propagation span much of the physical phenomena we experience and play a fundamental role both in engineered systems for communication and imaging as well as naturally-occurring aspects of earth's environment. Specific examples include: wave phenomena involved with natural disasters such as earthquakes and tsunamis; environmental irritants such as acoustic pollution near airports and in cities; electromagnetic phenomena of importance to defense and civilian applications, such as radar imaging; and applications in medicine such as the interaction of high-frequency ultrasound and tissue. This project will develop improved tools for simulating waves and will design associated general-purpose open-source software with the potential for significant impact in a range of important application areas.With the staggering increases in computational power that have been and continue to be achieved, we expect to simulate more difficult and comprehensive models of physical phenomena. For wave propagation problems posed in the time domain, this means problems with many wavelengths within the computational domain involving interactions with complex geometrical features. To treat such problems efficiently requires the use of high-order discretization methods to minimize the effects of dispersion and dissipation. This work will be focused on fundamental mathematical issues required for the further development of robust, high-order wave solvers. These include: i. Development and analysis of energy-stable high-order/high-resolution discretization methods on hybrid structured-unstructured grids. Specifically we will investigate coupling high-order upwind discontinuous Galerkin methods on unstructured grids near complex boundaries and material interfaces with more efficient structured grid methods such as novel spectral element methods based on Hermite-Birkhoff interpolation (also known as jet schemes) or upwind difference methods constructed from piecewise polynomial or band-limited interpolation functions. Both first-order and second-order hyperbolic systems will be considered. ii. Development, analysis, and implementation of hp-adaptive strategies for these methods. iii. Coupling with an open-source radiation boundary condition library (expected release late 2014) containing various implementations of complete radiation boundary conditions (CRBC). These allow a priori determination of the boundary condition parameters to guarantee any desired accuracy for isotropic, homogeneous models in the far field. iv. Leveraging the fact that CRBCs are stable for any Friedrichs system, extend their applicability to more complex physical models including anisotropy.