Mathematical Sciences: Computational Analysis of Multiple Scales Problems in Wave Propagation

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9600146 Hagstrom This project's primary focus is on the development, analysis and implementation of new and more efficient numerical methods for simulating wave propagation in the presence of multiple temporal and spatial scales. This includes the construction of efficient discretization methods and accurate radiation conditions for solving linear hyperbolic systems, the development of adaptive spectral methods for problems requiring highly accurate solutions, and the analysis of splitting schemes for the efficient integration of multiple time scales problems for partial differential equations. Specific physical applications to be considered include acoustic and electromagnetic wave propagation and the challenging problem of simulating dynamic combustion phenomena making use of detailed models of the physics. Basic analyses of the governing systems of nonlinear partial differential equations will also be undertaken. Multiple scales problems are difficult because the uniform resolution of the smallest scales present would lead to a prohibitively large computational problem, even for the fastest machines likely to be available in the coming decades. They are also a common feature of scientifically and technologically important phenomena in a number of distinct physical settings. They arise sometimes from the need to solve problems on large spatial domains or over long time periods, and other times from the appearance of very localized solution features, such as sharp wavefronts. In the latter case they are generally associated with singular perturbations of the governing equations. Singular perturbations may be thought of as small changes in the equations which can lead to large qualitative and quantitative changes in the solutions. Such problems can also often be analyzed by non- numerical asymptotic techniques and in this work asymptotic analysis is used in a number of ways to guide and complement numerical computations. The translation of theoretical advances into useful tools, and the consideration of more complicated and comprehensive mathematical models is emphasized so as to maximize the impact of the research on other science and engineering disciplines.