Stanley Osher

Uniformly high order accurate essentially non-oscillatory schemes, 111

We continue the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws. We present an hierarchy of uniformly high-order accurate schemes which generalizes Godunovs scheme and its second-order accurate MUSCL extension to an arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and, consequently, the resulting schemes are highly nonlinear.

High Frequency Electromagnetic Propagation/Scattering

Abstract : Two directions of work have been pursued under this effort. The first direction is concerned with further extensions of our geometric techniques for spatial and surface ray tracing and applications of these techniques to problems in high-frequency electromagnetics. In addition, high-frequency physical optics methods were developed for accurate and efficient calculation 0 scattered fields due to currents in the penumbra region. The second direction is concerned with development of grid methods for capturing the motion of self-intersecting interfaces and related PDEs connected with propagation of high-frequency waves. In both directions significant progress has been achieved.