On an adaptive monotonic convection—diffusion flux discretization scheme

Abstract

The present paper concerns numerical investigation of a two-dimensional steady convection—diffusion scalar equation. Specific attention is given to resolving spurious oscillations in a confined region of high gradient. In smooth regions, a high-order accurate solution is desired, while in regions containing a sharp gradient, the strategy of applying a monotonic capturing scheme is preferred. In this paper, we model fluxes by means of a finite element model which has spaces spanned by Legendre polynomials. The propensity to yield an irreducible diagonal-dominant stiffness matrix is an attribute of this finite element flux discretization scheme. Fundamental analyses are extensively conducted in this paper in two areas. First, by conducting a modified equation analysis, we are led to confirm the consistency of the scheme. Both the stability and monotonicity of the solutions are also addressed. A guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the M-matrix. This monotonicity study provides us with greater numerical insight into the importance of the chosen Peclet number. Beyond its critical value, oscillatory solutions are present in the flow. This monotonic region is, however, fairly restricted. It is this shortcoming which limits the finite element practioner's choices for fairly small grid size. The scope of application using the proposed upwind scheme is, thus, rather small because monotonic solutions are prohibitively expensive to compute. Circumvention of such deficiency is accomplished by making modifications to the structured grid. Several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients. Considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting.