Monotonicity Corrections for Nine-Point Scheme of Diffusion Equations (Journal of Computational Mathematics, 2024)
In this paper, we present a nonlinear correction technique to modify the nine-pointscheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resultedscheme preserves the positivity. We first express the flux by the cell-centered unknownsand edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinearcombination technique to get a monotone scheme. In order to obtain a cell-centered finitevolume scheme, we need to use the cell-centered unknowns to locally approximate theauxiliary unknowns. We present a new method to approximate the auxiliary unknownsby using the idea of an improved multi-points flux approximation. The numerical resultsshow that the new proposed scheme is robust, can handle some distorted grids that someexisting finite volume schemes could not handle, and has higher numerical accuracy thansome existing positivity-preserving finite volume schemes.
Keywords: Monotonicity corrections; Diffusion equation; Improved MPFA; Distorted meshes.