Spatial Discretization: The Diffusion Term

Abstract

This chapter describes in detail the discretization of the diffusion term represented by the spatial Laplacian operator. It is investigated separately from the convection term, because convection and diffusion represent two distinct physical phenomena. Thus from a numerical point of view, they have to be handled differently, requiring distinct interpolation profiles with disparate considerations. The chapter begins with the discretization of the diffusion equation in the presence of a source term over a two-dimensional rectangular domain using a Cartesian grid system. The adopted interpolation profile for the variation of the dependent variable between grid points and the basic rules that should be satisfied by the coefficients of the discretized equation are discussed. The chapter proceeds with a discussion on the implementation of the Dirichlet, Von Neumann, mixed, and symmetry boundary conditions. The discretization over a non-Cartesian orthogonal grid is then introduced, followed by a detailed description of the discretization on non-orthogonal structured and unstructured grid systems. The treatment of the non-orthogonal cross-diffusion contribution, which necessitates computation of the gradient, is clarified. Then anisotropic diffusion is introduced and handled following the same methodology developed for isotropic diffusion. The under-relaxation procedure needed for highly non-linear problems is outlined. The chapter ends with computational pointers explaining the treatment of diffusion in both uFVM and OpenFOAM^®.