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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 ϵ , 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 ϵ as a function of the local mesh size h j . In particular we show that choosing ϵ = h j 2 or ϵ = h j is beneficial for the error and convergence order, studying on several non-uniform 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 ϵ 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.

Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. In this paper we address this issue with a new Central WENOZ ( CWENOZ ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions (Arbogast et al. in SIAM J Numer Anal 56(3):1818-1947, 2018),(Balsara et al. in J Comput Phys 326:780-804, 2016). The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell.

In this paper we generalise to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy et al. (SIAM J Sci Comput 22(2):656–672, 2000 ), thus obtaining a truly two-dimensional non-oscillatory third order reconstruction with a very compact stencil and that does not involve mesh-dependent coefficients. This latter characteristic is quite valuable for its use in h-adaptive numerical schemes, since in such schemes the coefficients that depend on the disposition and sizes of the neighbouring cells (and that are present in many existing WENO-like reconstructions) would need to be recomputed after every mesh adaption. In the second part of the paper we propose a third order h-adaptive scheme with the above-mentioned reconstruction, an explicit third order TVD Runge–Kutta scheme and the entropy production error indicator proposed by Puppo and Semplice (Commun Comput Phys 10(5):1132–1160, 2011 ). After devising some heuristics on the choice of the parameters controlling the mesh adaption, we demonstrate with many numerical tests that the scheme can compute numerical solution whose error decays as ⟨ N ⟩ - 3 , where ⟨ N ⟩ is the average number of cells used during the computation, even in the presence of shock waves, by making a very effective use of h-adaptivity and the proposed third order reconstruction.

This paper is concerned with the construction of high order schemes on irregular grids for balance laws, including a discussion of an a-posteriori error indicator based on the numerical entropy production. We also impose well-balancing on non uniform grids for the shallow water equations, which can be extended similarly to other balance laws, obtaining schemes up to fourth order of accuracy with very weak assumptions on the regularity of the grid. Our results show the expected convergence rates, the correct propagation of shocks across grid discontinuities and demonstrate the improved resolution achieved with a locally refined non-uniform grid. The schemes proposed in this work naturally can also be applied to systems of conservation laws. They may also be extended to higher space dimensions by means of dimensional splitting. The error indicator based on the numerical entropy production, previously introduced for the case of systems of conservation laws, is extended to balance laws. Its decay rate and its ability to identify discontinuities is illustrated on several tests.

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 (CWENO\\mathsf {CWENO}). This technique relies on the same selection mechanism of smooth stencils adopted in WENO\\mathsf {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 hh-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 CWENO\\mathsf {CWENO} reconstruction studied here, for the same accuracy.

Central WENO reconstruction procedures have shown very good performance in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and higher 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 global smoothness indicator τ is still lacking. In this work we first prove results on the asymptotic expansion of one- and multidimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of CWENOZ schemes, which are in addition to 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 correctness of the analysis and the good performance of the new schemes.

Here we present the results of the inversion of a multi-annual temperature profile (2013, 2014, 2015) of the deepest borehole (235 m) in the mountain permafrost of the world located close to Stelvio Pass in the Central Italian Alps. The SHARE STELVIO Borehole (SSB) has been monitored since 2010 with 13 thermistors placed at different depths between 20 and 235 m. The negligible porosity of the rock (dolostone,  <  5 %) allows us to assume the latent heat effects are also negligible. The inversion model proposed here is based on the Tikhonov regularization applied to a discretized heat equation, accompanied by a novel regularizing penalty operator. The general pattern of ground surface temperatures (GSTs) reconstructed from SSB for the last 500 years is similar to the mean annual air temperature (MAAT) reconstructions for the European Alps. The main difference with respect to MAAT reconstructions relates to post Little Ice Age (LIA) events. Between 1940 and 1989, SSB data indicate a cooling of ca. 1 °C. Subsequently, a rapid and abrupt GST warming (more than 0.8 °C per decade) was recorded between 1990 and 2011. This warming is of the same magnitude as the increase in MAAT between 1990 and 2000 recorded in central Europe and roughly doubling the increase in MAAT in the Alps.

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 \\(\\epsilon\\), 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 \\(\\epsilon\\) as a function of the local mesh size \\(h_j\\). In particular we show that choosing \\(\\epsilon=h_j^2\\) or \\(\\epsilon=h_j\\) is beneficial for the error and convergence order, studying on several non-uniform 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 \\(\\epsilon\\) 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.

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.

We propose a high order numerical scheme for time-dependent first order Hamilton--Jacobi--Bellman equations. In particular we propose to combine a semi-Lagrangian scheme with a Central Weighted Non-Oscillatory reconstruction. We prove a convergence result in the case of state- and time-independent Hamiltonians. Numerical simulations are presented in space dimensions one and two, also for more general state- and time-dependent Hamiltonians, demonstrating superior performance in terms of CPU time gain compared with a semi-Lagrangian scheme coupled with Weighted Non-Oscillatory reconstructions.