Eventos Anais de eventos
COBEM 2021
26th International Congress of Mechanical Engineering
Numerical Simulation of the Diffusion Equation Via a Non-Linear Flux Splitting Technique with the Multipoint Flux Approximation Method with a Diamond Stencil Satisfying the Discrete Maximum Principle Using 2-D Unstructured Meshes
Submission Author:
ARTUR CASTIEL REIS DE SOUZA , PE , Brazil
Co-Authors:
ARTUR CASTIEL REIS DE SOUZA, Túlio de Moura Cavalcante, DARLAN KARLO ELISIÁRIO DE CARVALHO, Paulo Roberto Maciel Lyra
Presenter: ARTUR CASTIEL REIS DE SOUZA
doi://10.26678/ABCM.COBEM2021.COB2021-1660
Abstract
The locally conservative family of linear Control Volume Distributed Multipoint Flux Approximation (CVD-MPFA) has been successfully employed for decades to solve the diffusion equation in 2-D and 3-D domains. However, for highly anisotropic and heterogeneous media, these schemes may produce solutions with spurious oscillations in the scalar field that violate the Discrete Maximum Principle (DMP). To overcome this limitation, in this paper, we present a repair technique based on a new non-linear flux splitting approach for cell centered MPFA finite volume methods to solve the steady state diffusion problem with heterogeneous, possibly discontinuous, and anisotropic diffusion tensors. In particular, we implement our repair technique with the non-orthodox Multipoint Flux approximation Method with a Diamond Stencil (MPFA-D). Our technique is designed to eliminate spurious oscillations by imposing the DMP without losing mass conservation. This is done similarly to the M-Matrix Flux Splitting method that splits the Two-Point Flux Approximation (TPFA) contribution, which is naturally monotone, from the cross diffusion flux terms. We compute a parameter that calculates, for each control surface flux, the maximum amount of cross diffusion term at every iteration that makes the solution monotonic. The formulation is tested against a series of benchmark problems found in literature. Our results are compared with different linear and robust MPFA methods. For all examples solved, the non-linear flux-splitting technique has proven to be robust and accurate in a very efficient manner producing linear preserving (LP) solutions free of spurious oscillations honoring the DMP for arbitrary unstructured meshes and heterogeneous and anisotropic diffusion tensors.
Keywords
Heterogeneous and Anisotropic, Steady State Diffusion Equation, Media, Multipoint Flux Approximation with a Diamond Stencil (MPFA-D), Non-linear Flux-Splitting Scheme (NLFS), Discrete Maximum Principal (DMP)
