G. Puppo

C. Klingenberg, G. Puppo, M. Semplice

This work presents arbitrary high order well balanced finite volume schemes for the Euler equations with a prescribed gravitational field. It is assumed that the desired equilibrium solution is known, and we construct a scheme which is exactly well balanced for that particular equilibrium. The scheme is based on high order reconstructions of the fluctuations from equilibrium of density, momentum and pressure, and on a well balanced integration of the source terms, while no assumptions are needed on the numerical flux, beside consistency. This technique allows to construct well balanced methods also for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to non steady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in 2D.

I. Cravero, G. Puppo, M. Semplice et al.

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.