A linearity-preserving finite volume scheme with a diamond stencil for the simulation of anisotropic and highly heterogeneous diffusion problems using tetrahedral meshes

Abstract

In this paper, we propose a non-orthodox Multipoint Flux Approximation scheme with a “Diamond” stencil (MPFA-D) for the solution of the 3-D steady state diffusion equation. Following the work of GAO and WU (2011), in our method, the auxiliary vertex unknowns are eliminated by a novel explicit interpolation that is flux conservative and is constructed under the Linearity-Preserving Criterion (LPC). The MPFA-D is able to reproduce piecewise linear solutions exactly on challenging heterogeneous and anisotropic media, even in cases with some severely distorted meshes. Furthermore, our new scheme presentssecond order accuracy for the scalar unknown and, at least, first order accuracy for fluxes, considering unstructured tetrahedral meshes and arbitrarily anisotropic diffusion tensors. In order to validate our numerical scheme, we perform different test cases, involving 3-D benchmarks on diffusion problems. We compare the performance with other schemes found in literature. We also compare our Linearity-Preserving Explicit Weight (LPEW) interpolation with other interpolations strategies to evaluate its robustness to handle anisotropic and heterogeneous, possibly discontinuous diffusion tensors. In general, our linear preserving MPFA-D method performs well, however it is not monotone, particularly for very distorted meshes and highly anisotropic diffusion tensors.