Robust and accurate viscous discretization via upwind scheme - I: Basic principle

In this paper, we introduce a general principle for constructing robust and accurate viscous discretization, which is applicable to various discretization methods, including finite-volume, residual-distribution, discontinuous-Galerkin, and spectral-volume methods. The principle is based on a hyperbolic model for the viscous term. It is to discretize the hyperbolic system by an advection scheme, and then derive a viscous discretization from the result. A distinguished feature of the proposed principle is that it automatically introduces a damping term into the resulting viscous scheme, which is essential for effective high-frequency error damping and, in some cases, for consistency also. In this paper, we demonstrate the general principle for the diffusion equation on uniform grids in one dimension and unstructured grids in two dimensions, for node/cell-centered finite-volume, residual-distribution, discontinuous-Galerkin, and spectral-volume methods. Numerical results are presented to verify the accuracy of the derived diffusion schemes and to illustrate the importance of the damping term for highly-skewed typical viscous grids. (C) 2011 Elsevier Ltd. All rights reserved.