Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 1: 143 - 169

DOI: https://doi.org/10.1007/s42967-021-00151-4

ORIGINAL PAPERS

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

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  • 1 Dipartimento di Scienza e Alta Tecnologia, Università dell'Insubria, Via Valleggio, 11, 22100 Como, Italy;
    2 Dipartimento di Matematica, Università di Torino, Via C. Alberto, 10, 10124 Torino, Italy;
    3 Dipartimento di Matematica, Università La Sapienza, P. le Aldo Moro, 5, 00185 Roma, Italy

Received date: 2021-01-31

  Revised date: 2021-06-03

  Online published: 2023-03-08

Supported by

This work was funded by MIUR-PRIN project 2017KKJP4X “Innovative numerical methods for evolutionary partial differential equations and applications”. Gabriella Puppo acknowledges also the support of 2019 Ateneo Sapienza research project no. RM11916B51CD40E1.

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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 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. org/ 10. 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.

Key words： High-order finite volume schemes; Boundary conditions without ghost cells; Hyperbolic systems; CWENOZ reconstruction; Adaptive order reconstructions

Cite this article

M. Semplice, E. Travaglia, G. Puppo . One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells[J]. Communications on Applied Mathematics and Computation, 2023 , 5(1) : 143 -169 . DOI: 10.1007/s42967-021-00151-4

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