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

TL;DR

This paper introduces improved one- and multi-dimensional CWENOZ reconstructions that avoid ghost cells for boundary condition implementation in finite volume schemes, enhancing accuracy near domain boundaries.

Contribution

It extends a ghost-cell-free boundary reconstruction technique to higher dimensions and provides more accurate methods for third order finite volume schemes.

Findings

The new reconstructions outperform standard ghost cell methods in accuracy.

Numerical tests confirm improved boundary treatment without ghost cells.

The methods are applicable to Cartesian grids in multiple dimensions.

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.