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

Improved averaging-based finite-difference schemes for the convection-diffusion equation

Convection and diffusion have significant effect on reservoir flow systems. To represent the reservoir properly, these effects are encountered in models of reservoir engineering. Many researchers have worked on the convection-diffusion equations. Barakat and Clark considered a diffusion equation in 1966 and proposed a new explicit finite-difference scheme which gives a better result with less error using an averaging of two lower-order schemes. Bokhari and Islam extended this work to solve convection-diffusion equations and claimed to get the accuracy o