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The weighted essentially-oscillatory (WENO) schemes are a class of finite volume and finite difference methods for solving convection-dominated problems, mainly hyperbolic conservation laws. The idea comes from the earlier essentially-oscillatory (ENO) schemes, first developed in [1] in finite volume version and in [6] in finite difference version, for solving hyperbolic conservation laws. After the initial appearance of the WENO schemes in [3] and [2], they have been under continuous development and applications. Since the WENO schemes are very robust and stable for solving nonlinear problems containing strong shocks, as well as are able to maintain high order accuracy in smooth regions, they have become a very popular class of numerical methods for solving application problems with both discontinuities or sharp gradient regions and complex smooth region structures. The lecture notes [4] and the recent review paper [5] are good references for knowing more about the WENO schemes.
In this CAMC focused issue on WENO schemes, we have 16 papers covering a wide spectrum, from algorithm formulation, analysis to implementation and applications. We hope this focused issue will serve as a focused reference source for recent development of the WENO schemes, and will help to stimulate more research and wider applications of the WENO schemes in the future.
References
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Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E., Quarteroni, A. (eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)
Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
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Gottlieb, S., Hesthaven, J.S., Qiu, J. et al. Preface to the Focused Issue on WENO Schemes. Commun. Appl. Math. Comput. 5, 1–2 (2023). https://doi.org/10.1007/s42967-022-00196-z
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DOI: https://doi.org/10.1007/s42967-022-00196-z