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

Abstract

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.

References in corpus (8)

• On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes
• CWENO: uniformly accurate reconstructions for balance laws
• Optimal definition of the nonlinear weights in multidimensional Central WENOZ reconstructions
• Third and fourth order well-balanced schemes for the shallow water equations based on the CWENO reconstruction
• Central WENO schemes through a global average weight
• High Order Weighted Extrapolation for Boundary Conditions for Finite Difference Methods on Complex Domains with Cartesian Meshes
• On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws
• Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws