CWENO: uniformly accurate reconstructions for balance laws
For computational scientists using finite volume schemes, CWENO provides a more efficient reconstruction method for balance laws and adaptive mesh refinement, though it is an incremental improvement over existing WENO/ENO techniques.
The paper introduces CWENO, a framework for high-order essentially non-oscillatory reconstructions that uses a central stencil and polynomials of varying degrees, enabling uniform accuracy within a cell. This method is particularly effective for balance laws, h-adaptive schemes, and moving mesh techniques, offering improved efficiency over ENO methods by requiring a narrower stencil for the same accuracy.
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.