New Methods for the Simulation and Analysis of Waves

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New Methods for the Simulation and Analysis of Waves Thomas Hagstrom 9971772We will develop, analyze, implement and apply new methods for the solutionof wave propagation problems. Our primary focus is on numerical techniques which can efficiently and reliably utilize the capabilities of modern computers to simulate problems whose scale has hitherto precluded their detailed study. A large scale wave propagation problem will generally be given in an extended spatial and/or temporal domain. In engineering applications, the domain will also usually involve bodies with complex shapes. These features all pose difficult challenges to numerical algorithms. The radiation of energy to the far field is a feature of most wave systems and is associated with large spatial domains. To make computational ssolutions feasible, the domain must be artificially truncated. This truncation introduces errors which have been difficult to estimate or reduce. In recent years, we, along with other researchers, have developed new techniques for imposing accurate truncations, which solve this problem in some special but important cases. As part of this project, we will extend the applicability of these new methods. Long time simulations also make special demands on the accuracy of numerical approximations. In particular, small inaccuracies in the wave speeds or the effects of numerical damping will generally accumulate to produce large errors. Therefore, highly accurate methods must be used. However, these are typically difficult to apply in complex geometries. We will work on the development and analysis of new methods with high orders of accuracy which can be more easily utilized near bodies with complicated shapes. Finally, we will study some basicproblems related to the existence, smoothness, and asymptotic approximability of solutions to the compressible Navier-Stokes equations, which describe the motion of most common liquids and gases. We are particularly interested in flows which are slow in comparison with the speed of sound. Such flows are typically modeled by the incompressible Navier-Stokes equations, whose mathematical theory, though still quite incomplete, is better developed. We, however, retain the compressible effects, and study both theoretically and by simulation the relationship between the solutions of the two systems. In our work we focus on some specific physical systems, primarily from thefields of acoustics and fluid dynamics. As such, we hope to enhance our capability to use high-performance computing to predict the production andinteraction of sound with fluid flows. However, due to the general importance of wave theory and the underlying unity of its mathematical description, most of our results will be directly applicable in diverse areas such as electromagnetism and elasticity. In the long term, an improved capability to simulate waves will have important applications including the reduction of aircraft noise, the improvement of radar and sonar imaging, the predictabilityof the effects of earthquakes, and the design of better communications systems.