High Order Mimetic Differences Applied to the Convection–diffusion Equation: a Matrix Stability Analysis

The convection–diffusion equation describes physical phenomena where particles or energy are transferred within a physical system due to the processes of diffusion and convection. In this work, we investigate a discretization framework based on the mimetic fourth-order finite-difference staggered-grid Castillo–Grone (CG) operators (Castillo et al. in Appl Numer Math 37(1–2):171–187, 2001. https://doi.org/10.1016/S0168-9274(00)00033-7 ; Castillo and Grone in SIAM J Matrix Anal Appl 25(1):128–142, 2003. https://doi.org/10.1137/S0895479801398025 ), which have a sextuple of free parameters. We study the dependency of the stability and precision properties of our numerical scheme based on these CG free parameters, and propose parameters that favor both properties. We compare our results with CG parameters previously mentioned in the literature, including those leading to mimetic operators of minimum bandwidth.