A least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes

Abstract

We present a new least squares based diamond scheme for anisotropic diffusion problems on polygonal meshes. This scheme introduces both cell-centered unknowns and vertex unknowns. The vertex unknowns are intermediate ones and are expressed as linear combinations with the surrounding cell-centered unknowns by a new vertex interpolation algorithm which is also derived in least squares approach. Both the new scheme and the vertex interpolation algorithm are applicable to diffusion problems with arbitrary diffusion tensors and do not depend on the location of discontinuity. Benefitting from the flexibility of least squares approach, the new scheme and vertex interpolation algorithm can also be extended to 3D cases naturally. The new scheme and vertex interpolation algorithm are linearity-preserving under given assumptions and the numerical results show that they maintain nearly optimal convergence rates for both L 2 error and H 1 error in general cases. More interesting is that a very robust performance of the new vertex interpolation algorithm on random meshes compared with the algorithm LPEW2 is found from the numerical tests. K E Y W O R D Scell-centered scheme, diffusion problem, least squares approach, linearity-preserving, vertex interpolation INTRODUCTIONAnisotropic diffusion problems arise in a great many applications, such as radiation hydrodynamics(RHD), magnetohydrodynamics(MHD), plasma physics, reservoir modeling, and so on. The finite volume (FV) method is one of the most popular methods in the simulation of diffusion processes because of the simplicity, local conservation and some other nice numerical properties. In the recent decades, there has been extensive research on designing efficient FV schemes for anisotropic diffusion problems on general meshes. The readers are referred to References 1-4 and the references cited therein for recent developments. Here we mention the cell-centered FV scheme early proposed in Reference 5 and studied by many works, such as Reference 6-8. It is called diamond scheme because of the diamond shaped stencil of the flux approximation. Since the resulting FV equation has a nine-point stencil on the structured quadrilateral meshes, it is also sometimes called nine point scheme. Generally the diamond scheme introduces both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and finally eliminated by the linear combinations of the surrounding cell-centered ones to get a pure cell-centered scheme. The vertex interpolation algorithm is the key ingredient for the diamond scheme to achieve optimal accuracy.