CWENO: uniformly accurate reconstructions for balance laws

TL;DR

This paper introduces CWENO, a high-order, uniformly accurate reconstruction method for balance laws that uses a compact stencil and polynomial selection, improving upon traditional ENO techniques for finite volume schemes.

Contribution

The paper presents CWENO, a novel reconstruction framework that achieves high-order accuracy with a compact stencil, enabling uniform function approximation for balance laws and improving over ENO methods.

Findings

Provides an analytic polynomial interpolant for uniform approximation.

Reduces stencil width compared to ENO for same accuracy.

Enhances high-order quadrature and adaptive mesh refinement applications.

Abstract

In this paper we introduce a general framework for defining and studying essentially non-oscillatory reconstruction procedures of arbitrarily high order accuracy, interpolating data in a central stencil around a given computational cell ($\CWENO$). This technique relies on the same selection mechanism of smooth stencils adopted in $\WENO$, but here the pool of candidates for the selection includes polynomials of different degrees. This seemingly minor difference allows to compute an 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 then the $\CWENO$ reconstruction studied here, for the same accuracy.