TY - JOUR
T1 - An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes
AU - Dong , Cheng
AU - Kang , Tong
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 405
EP - 437
PY - 2021
DA - 2021/01
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2020-0006
UR - https://global-sci.org/intro/article_detail/nmtma/18605.html
KW - Cell-centered scheme, nonlinear diffusion equation, vertex interpolation algorithm,
linearity-preserving criterion, numerical heat-barrier issue, finite volume scheme.
AB - <p style="text-align: justify;">In this paper, we suggest a new vertex interpolation algorithm to improve
an existing cell-centered finite volume scheme for nonlinear diffusion problems on
general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method
for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability
of the corresponding local system is proved under some assumptions. Additionally,
we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical
heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The
final scheme allows arbitrary continuous or discontinuous diffusion coefficients and
can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux
are observed in numerical experiments. In the comparative experiments with some
existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.</p>