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CWENO: Uniformly accurate reconstructions for balance laws


Authors: I. Cravero, G. Puppo, M. Semplice and G. Visconti
Journal: Math. Comp. 87 (2018), 1689-1719
MSC (2010): Primary 65M08; Secondary 65M12
DOI: https://doi.org/10.1090/mcom/3273
Published electronically: November 2, 2017
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Abstract: In this paper we introduce a general framework for defining and studying essentially nonoscillatory reconstruction procedures of arbitrarily high order of accuracy, interpolating data in the central stencil around a given computational cell ( $ \mathsf {CWENO}$). This technique relies on the same selection mechanism of smooth stencils adopted in $ \mathsf {WENO}$, but here the pool of candidates for the selection includes polynomials of different degrees. This seemingly minor difference allows us to compute the analytic expression of a polynomial interpolant, approximating the unknown function uniformly within a cell, instead of only at one point at a time. For this reason this technique is particularly suited for balance laws for finite volume schemes, when averages of source terms require high order quadrature rules based on several points; in the computation of local averages, during refinement in $ h$-adaptive schemes; or in the transfer of the solution between grids in moving mesh techniques, and in general when a globally defined reconstruction is needed. Previously, these needs have been satisfied mostly by ENO reconstruction techniques, which, however, require a much wider stencil than the $ \mathsf {CWENO}$ reconstruction studied here, for the same accuracy.


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Additional Information

I. Cravero
Affiliation: Dipartimento di Matematica, Università di Torino, Via C. Alberto, 10, Torino, Italy
Email: isabella.cravero@unito.it

G. Puppo
Affiliation: Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio, 11, Como, Italy.
Email: gabriella.puppo@uninsubria.it

M. Semplice
Affiliation: Dipartimento di Matematica, Università di Torino, Via C. Alberto, 10, Torino, Italy
Email: matteo.semplice@unito.it

G. Visconti
Affiliation: Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio, 11, Como, Italy
Address at time of publication: IGPM–RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany
Email: visconti@igpm.rwth-aachen.de

DOI: https://doi.org/10.1090/mcom/3273
Keywords: High order accuracy, essentially nonoscillatory, finite volume schemes, balance laws, nonuniform grids.
Received by editor(s): July 26, 2016
Received by editor(s) in revised form: March 11, 2017
Published electronically: November 2, 2017
Additional Notes: The third author is the corresponding author.
This work was supported by INDAM GNCS-2016 grant “Metodi numerici per la quantificazione dellâincertezza in equazioni iperboliche e cinetiche”
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society