On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes

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

Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction

In this paper we generalize to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput., 2001), 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 neighboring 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., 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 $\langle N\rangle^{-3}$, where $\langle N\rangle$ 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.Comment: many updates to text and figure

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in the boundary cells. In Naumann et al. (Appl. Math. Comput. 325: 252-270. https://doi.org110.1016/j.amc.2017.12.041, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests, we compare the novel reconstruction with the standard approach using ghost cells

Quinpi: Integrating Conservation Laws with CWENO Implicit Methods

Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first-order schemes. High order schemes instead also need to control spurious oscillations, which requires limiting in space and time also in the linear case. We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear weights of a standard high order space reconstruction, and to achieve limiting in time. In this preliminary work, we concentrate on the case of a third-order scheme, based on diagonally implicit Runge Kutta (DIRK) integration in time and central weighted essentially non-oscillatory (CWENO) reconstruction in space. The numerical tests involve linear and nonlinear scalar conservation laws

One- and multi-dimensional CWENOZ reconstructions for implementing boundary conditions without ghost cells

We address the issue of point value reconstructions from cell averages in the context of third order finite volume schemes, focusing in particular on the cells close to the boundaries of the domain. In fact, most techniques known in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that, taking into account the boundary conditions, fills the ghost cells with appropriate values, so that a standard reconstruction can be applied also in boundary cells. In (Naumann, Kolb, Semplice, 2018), motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network, a different technique was explored that avoids the use of ghost cells, but instead employs for the boundary cells a different stencil, biased towards the interior of the domain. In this paper, extending that approach, which does not make use of ghost cells, we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids. In several numerical tests we compare the novel reconstruction with the standard approach using ghost cells

ARBITRARY ORDER FINITE VOLUME WELL-BALANCED SCHEMES FOR THE EULER EQUATIONS WITH GRAVITY

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, velocity, 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 also allows one to construct well-balanced methods for a class of moving equilibria. Several numerical tests demonstrate the performance of the scheme on different scenarios, from equilibrium solutions to nonsteady problems involving shocks. The numerical tests are carried out with methods up to fifth order in one dimension, and third order accuracy in two dimensions

Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations

Abstract. In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computa-tional cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its con-vergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution