CWENO: Uniformly accurate reconstructions for balance laws

Cravero, I.; Puppo, G.; Semplice, M.; Visconti, G.

doi:10.1090/mcom/3273

CWENO: Uniformly accurate reconstructions for balance laws
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by I. Cravero, G. Puppo, M. Semplice and G. Visconti PDF

Math. Comp. 87 (2018), 1689-1719 Request permission

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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