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.MSC 65M08, 65M12.

Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called , that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing as a function of the local mesh size hj. In particular we show that choosing = h 2 j or = hj is beneficial for the error and convergence order, studying on several nonuniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.

Central WENO reconstruction procedures have shown very good performances in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and more space dimensions, on different types of meshes. Their most recent formulations include WENOZ-type nonlinear weights, but in this context a thorough analysis of the definition of the global smoothness indicator τ is still lacking. In this work we first prove results on the asymptotic expansion of multi-dimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of CWENOZ schemes also beyond those considered in this paper. Next, we introduce the optimal definition of τ for the one-dimensional CWENOZ schemes and for one example of two-dimensional CWENOZ reconstruction. Numerical experiments of one and two dimensional test problems show the good performance of the new schemes.Keywords. Central WENOZ (CWENOZ) essentially non-oscillatory reconstructions finite volume schemes smoothness indicators MSC2010. 65M08 65M20In this section we prove the theoretical results to analyse the CWENO and CWENOZ reconstruction procedures in one and more space dimensions. To this end, we restrict here to the scalar case, since usually, in the case of systems of conservation laws, the reconstruction procedures are applied component-wise, directly to the conserved variables or after the local characteristic projection.

This work is dedicated to the development and comparison of WENO-type reconstructions for hyperbolic systems of balance laws. We are particularly interested in high order shock capturing non-oscillatory schemes with uniform accuracy within each cell and low spurious effects. As a tool to measure the artifacts introduced by a numerical scheme, we study the deformation of a single Fourier mode and introduce the notion of distorsive errors, which measure the amplitude of the spurious modes created by a discrete derivative operator. Further we refine this notion with the idea of temperature, in which the amplitude of the spurious modes is weighted with its distance in frequency space from the exact mode. In this analysis, linear schemes have zero temperature. Of course, in order to prevent oscillations it is necessary to introduce nonlinearities in the scheme, thus increasing their temperature. However it is important to warm up the linear scheme just enough to prevent spurious oscillations. With several tests we show that the newly introduced CWENOZ schemes are cooler than other existing WENO-type operators, while maintaining good non-oscillatory properties

We propose a wavelet-based procedure for adapting a finite element mesh to the structure of the solution. After a finite element solution is computed on a given unstructured mesh, it is wavelet-analyzed on a superimposed regular dyadic grid; the analysis leads to an adapted distribution of grid points, which defines the new unstructured mesh via a Delaunay triangulation. Several examples of discretizations of steady convection-diffusion problems in the convection-dominated regime indicate the feasibility of our approach.

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In this paper we consider and analyse NURBS based on bivariate quadratic B-splines on criss-cross triangulations of the parametric domain Ω0 = [0, 1] × [0, 1], presenting their main properties, showing their performances to exactly construct quadric surfaces and reporting some applications related to the modeling of objects. Moreover, we propose applications to the numerical solution of partial differential equations, with mixed boundary conditions on a given physical domain Ω, by using three different spline methods to set the prescribed Dirichlet boundary conditions.

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