Robust and Efficient Numerical Methods for Wave Equations in the Time Domain: Nonlinear and Multiscale Problems

Southern Methodist University, Dallas TX

Investigators

Abstract

The project aims to develop fast and reliable algorithms for simulating waves, capable of exploiting current and future computing technology and of treating a wide range of mathematical models of physical systems. Wave phenomena are important in almost all areas of the physical sciences and engineering. They are central to sensing and imaging applications, to the modeling of potential natural disasters such as earthquakes, and to humanity's quest to understand the universe. In the longer term, the algorithms developed will be incorporated into software to promote their broad use. Besides contributions to basic tools for solving comprehensive models of wave physics, the project will result in the training of three graduate students in computational science, supported in part as funded research assistants, and in the further development of transdisciplinary programs in research and education which involve computation as an important component. The main challenges to computational technique posed by problems in wave theory are rooted in the multiple temporal and spatial scales which typically occur. The fundamental fact that a wave propagates over many wavelengths without significant attenuation leads to the requirement that numerical methods must have minimal dispersion and dissipation errors and that the computational domain must be limited using accurate approximate radiation boundary conditions. In addition, the waves may interact with complex geometrical objects with their own inherent length scales. The project will address all of these issues. Building on previous success in developing optimal rational approximations to radiation boundary conditions for linear hyperbolic problems in uniform media, reduced order modeling algorithms will be used to construct effective methods for problems with Coulomb potentials, in periodic media, in multiscale media, and for Schrodinger-type equations. The investigator will improve the efficiency of their discretization methods, which are robust, due to energy stability, have arbitrary convergence orders for smooth solutions, and are capable of treating general second-order hyperbolic systems arising from action principles. Specifically, the research will develop higher order local time-stepping schemes to more efficiently treat locally refined and hybrid meshes and will exercise their methods on interesting physical problems, including gravitational and topological waves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.