High Order Methods for Linear and Nonlinear Waves

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Shu0207451 The investigator and his colleague study high-order-accuracycomputational methods for linear and nonlinear waves, withemphasis on shock wave calculations and simulation ofelectro-magnetics waves. The work includes the development andanalysis of high-order finite difference, finite element, andspectral methods, as well as applications of these methods tocomputational fluid dynamics and computational electro-magnetics.In particular, the efforts include the following components:high-order methods for shock wave calculations, including finitedifference WENO schemes, spectral methods for supersonic reactiveflows, and finite element discontinuous Galerkin methods;high-order methods for Maxwell's equations; and perfectly matchedabsorbing layers. Computers are now used more extensively in engineering andother applied sciences. An important component to effectivelyuse computers is the design and analysis of efficient numericalalgorithms. Important applications such as computer-aided designof aircraft are to a large extent dependent on efficient andreliable algorithms designed by applied mathematicians. Thehigh-order methods the investigators develop and analyze in thisproject should significantly increase the efficiency andreliability of algorithms used in crucial application areas suchas aerospace industry, communications, and material science.