Abstract
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 Godunov’s 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.
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Dedicated to Peter Lax on his 60th Birthday
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© 1987 Academic Press, Inc.
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Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R. (1987). Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III. In: Hussaini, M.Y., van Leer, B., Van Rosendale, J. (eds) Upwind and High-Resolution Schemes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60543-7_12
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DOI: https://doi.org/10.1007/978-3-642-60543-7_12
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