Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes

Abstract

In this paper, we introduce a general principle for constructing time-accurate diffusion schemes, which is applicable to various discretization methods, includingnite-volume, residual- distribution, discontinuous-Galerkin, and spectral-volume methods. The principle is based on a hyperbolic relaxation-system model for diffusion. It is to discretize the hyperbolic system by an advection scheme, and then derive a diffusion scheme by bringing it to the equilibrium limit in a rather simple manner. A distinguished feature of the proposed principle is that it automatically introduces a damping term into the derived diffusion scheme, which is essential for effective high-frequency error damping and, in some cases, for consistency also. Another useful feature is that the derived diffusion scheme has the same implementation structure as a corresponding advection scheme, which makes it remarkably simple to integrate it with the advection scheme for advection-diffusion problems. We demonstrate the general principle by constructing diffusion schemes on uniform grids in one dimension and unstructured grids in two dimensions, for node/cell-centerednite-volume, residual-distribution, discontinuous- Galerkin, and spectral-volume methods. Numerical results are presented to verify the accuracy of the diffusion schemes and to illustrate the importance of the damping term. It is also shown that derived diffusion schemes yield comparably or more accurate solutions than widely-used schemes for time-dependent diffusion problems on isotropic/anisotropic irregular triangular grids.