Numerical Methods for Wave Propagation Problems: Efficient Resolution of Multiple Scales

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The focus of our research will be the detailed study of questions we deem crucial to the development of reliable, efficient, and general computational tools for wave propagation problems. These, in turn, can have long-term impacts on numerous fields in science and engineering. Precisely we will:(i) Further develop accurate methods for truncating the computational domain near regions where full approximations are required, extending the range of application of the successful methods we have previously constructed to inhomogeneous and anisotropic media as well as to multiscale computations; (ii) Construct and analyze novel high-resolution approximation schemes enabling accurate simulations with near-optimal degrees-of-freedom per wavelength, mild time-step stability restrictions, and easy coupling with grid grid generation software to efficiently treat problems in complex geometry; (iii) Apply our methods to difficult problems in aeroacoustics; (iv) Collaborate with other computational scientists who are building andmaintaining high-quality software for simulating waves. Wave propagation phenomena are ubiquitous in nature. Although waves may be produced by physical processes ranging from electric currents to turbulent flows to massive earthquakes, their basic features allow a unified mathematical description. From the perspective of simulations on modern computers, it is reasonable to hope that generally applicable tools can be constructed which will be useful in answering important questions throughout the basic and applied sciences. The challenge in the computational analysis of waves is that almost all problems of interest exhibit widely varying spatial scales. This is a consequence of the fundamental fact that waves propagate long distances relative to their characteristic dimension, the wavelength. We thus will work to develop methods which allow us to avoid the direct computation of the wave field everywhere along its path, concentrating computational resources only where they are needed. In addition to our work on basic techniques with broad applications, we plan focused studies on problems related to the generation of sound by jets and its propagation into the environment. We believe that the fundamental studies we will carry out can motivate the development of better sound suppression technologies for commercial and military aircraft.