Comparisons of compact and classical finite difference solutions of stiff problems on nonuniform grids

Abstract

In this paper we address the applicability of fourth-order and second-order finite difference schemes to problems which admit solutions with sharp gradients over thin layers. An optimum finite difference scheme is sought based on a thorough study of the convergence and accuracy properties of (1) classical second-order and fourth-order; (2) compact fourth-order; and (3) mixed second/fourth-order finite difference schemes. The four mixed-order finite difference schemes considered result from approximating the first and second derivatives within the nonuniform grid regions by the use of either compact or classical fourth-order and second-order finite difference schemes. The regions where severe gradients are anticipated are collocated with grid points that are distributed according to geometric progressions. Two representative boundary layer problems in fluid dynamics are chosen for this study, one governed by the Burgers equation and the other by the Reynolds equation. The convergence properties of the finite difference schemes are limited by the appearance of numerical oscillations at the grid interfaces between the uniform and nonuniform subdomains. The abrupt variation in the convergence rates corresponds to the transition from oscillatory solutions to oscillation-free solutions. It is shown that the oscillations are efficiently eliminated by the use of appropriate progression ratios in the nonuniform subdomains. For constant progression ratios, the results show that all of the considered finite difference methods, both lower-order classical schemes and higher-order compact schemes, exhibit qualitatively the same rates of convergence with respect to the uniform grid spacing. However, the compact scheme achieves vastly superior accuracies, but requires higher nonuniform grid ratios, and hence more grid points to resolve the oscillations. As the numerically generated oscillations at the interface are mitigated and eliminated on an appropriate nonuniform computational grid, the compact method is shown to be superior to classical second-order and fourth-order mixed methods in accuracy and computational efficiency for problems with boundary or interior layers.