Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes (Communications in Computational Physics, 2021)
In this article we present two types of nonlinear positivity-preserving finitevolume (PPFV) schemes for a class of three-dimensional heat conduction equations ongeneral polyhedral meshes. First, we present a new parameter selection strategy on theone-sided flux and establish a nonlinear PPFV scheme based on a two-point flux withhigher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai,H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes ourinterpolation formulae suitable to be constructed by existing approaches with highaccuracy and well robustness (e.g., the finite element method), thus enhancing theadaptability to distorted meshes with large deformations. Then we derive a linearmulti-point flux involving combination coefficients and, via the Patankar trick, obtainanother nonlinear PPFV scheme that is concise and easy to implement. The selectionstrategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preservingproperties of these two nonlinear PPFV solutions are proved. Numerical experimentswith the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that ourschemes both can achieve ideal-order accuracy even on severely distorted meshes.