CWENO: Uniformly accurate reconstructions for balance laws

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 ( C W E N O CWENO ). This technique relies on the same selection mechanism of smooth stencils adopted in W E N O 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 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 C W E N O CWENO reconstruction studied here, for the same accuracy.