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Abstract.In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation.Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries.These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.

DEDICATED TO PETER LAX ON HIS 60TH BIRTHDAYWe 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 Godunov's scheme and its secondorder 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 ovei 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.

In situations where the computed solution approximates a steady solution which possesses a regular far field asymptotic expansion, coordinate transformations are a traditional and an effective way to eliminate the difficulties with computational boundaries. This is certainly the case for many of the steady state problems in transonic flow. On the other hand, when highly oscillatory transient solutions are computed or when the computed solution has an asymptotic expansion with essential singularities near infinity, coordinate mapping techniques are ineffective and alternate approaches must be developed for treating the computational boundaries. These circumstances are typical for most calculations in seismology and some of the calculations in unsteady transonic flow and meterology.

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.