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Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. (English) Zbl 0927.65111
Quarteroni, Alfio (ed.) et al., Advanced numerical approximation of nonlinear hyperbolic equations. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 23–28, 1997. Berlin: Springer. Lect. Notes Math. 1697, 325-432 (1998).
Summary: We describe the construction, analysis, and application of ENO (essentially non-oscillatory) and WENO (weighted essentially non-oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high-order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities.
The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.
These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code them up for applications. Sample codes are also available from the author.
For the entire collection see [Zbl 0904.00047].

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis