Numerical Methods for Multiple Scale Problems in Wave Propagation: Efficient Approximation of Integral Operators in the Time Domain

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Many of the main obstacles to the development of efficient and reliable computational tools for simulating waves are rooted in the multiple spatial scales which are universally present. The focus of this project is the detailed study of select questions which are relevant for overcoming these obstacles. A unifying feature of the questions addressed is that all involve the efficient approximate evaluation of integral operators in space and time. Although the work, if successful, has the potential to impact numerous scientific and engineering disciplines, the efforts will be directed towards problems in aeroacoustics and electromagnetics. Precisely, the following will be developed: (i) Accurate, efficient and reliable computational domain truncation methods, allowing the direct simulations to take place only in regions where the medium is complex or where nonlinear effects are important; (ii) Efficient time-stepping methods allowing the simple treatment of concentrated regions of high resolution or geometric detail.Wave propagation problems are of fundamental importance in many areas of applied science and technology. They encompass a wide range of physics (electromagnetics, fluid and solid mechanics), but share essential mathematical properties. The defining characteristic of a wave is its ability to travel long distances relative to its basic dimension, the wavelength, carrying detailed information about the medium through which it has traveled. For this reason, waves are the primary method by which we probe nature and communicate. A consequence of this fundamental characteristic is that wave propagation problems typically involve disparate spatial scales - from the geometrical details of scatterers through a range of wavelengths to the propagation distances. These multiple scales, in turn, lead to difficulties in computational analysis. In particular, their uniform resolution would lead to a prohibitive number of degrees of freedom. Thus methods must be developed which can concentrate computational resources only where they are needed, providing the primary motivation for the problems we consider. In addition to this analysis, the plan is to collaborate with researchers who are building software for simulating jet noise, electromagnetic scattering in complex structures, as well as general-purpose wave propagation problems. Thus any positive developments from the research can come into use as rapidly as possible.