High-Order Embedded Interface Methods for Wave-Problems

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We propose to develop a new family of stable high-order finitedifference methods suitable for the solution of wave problemsinvolving many interfaces and significant geometric complexity.These complications are addressed by embedding the computationalproblem into a Cartesian grid and formulating methods such thatthe position of the material interfaces as well as the physicalproperties of the solution across the interface is accounted forproperly to the order of the scheme. Staggered grid as well asnon-staggered grid methods will be explored with the emphasis onthe development of a rigorous mathematical foundation for theseschemes to ensure robustness and uniform stability for all grid sizes and embedded geometries. Appealing properties of embeddingmethods such as the ability to model moving interfaces and theintroduction of virtual interfaces to enhance parallel performancewill be exploited to model a variety of wave problems inelectromagnetics, acoustics, seismology, and elasticity.The increasing interest in the accurate and efficient solutionof wave-dominated problems, e.g. problems in acoustic and electromagnetics, over very long periods of time has spawned an interest in the formulation of high-order accurate computationalmethods for such problems. In this effort we propose to developa new class of computational techniques, specifically aimed atthe reliable and robust modeling of problems of a realisticsize and complexity, e.g., the modeling of the propagation ofelectromagnetic and acoustic noise and its environmental impact, andthe modeling of underground waves of interest to the oil industry.The proposed methods are unique in maintaining a very simple computational structure without sacrificing the accuracy, henceovercoming a number of well known difficulties associated withexisting methods which require some kind of automated generation ofthe computational grid on which the solution is computed.These proposed developments will enable the modeling ofvery complex and realistic scenarios and will, in combination with high-performance computing facilities for which themethods are well suited, allow for the modeling and analysisof complex wave dominated problems in a variety of areas of interestto engineers and scientists.