A study of multidomain compact finite difference schemes for stiff problems

In this paper, we investigate the applicability of fourth-order and second-order finite difference schemes to problems which admit nonsingular, thin boundary or interior layers. An optimum finite difference scheme is sought based on a thorough study of the convergence and accuracy properties of the classical second-order, and classical fourth-order, compact fourth-order, and mixed second/fourth-order finite difference schemes. The mixed-order finite difference schemes considered result from approximating the first and second derivatives within the nonuniform grid subdomains by the use of either compact or classical fourth-order and second-order schemes. The computational domain is divided into subdomains which are refined independently according to the stiffness of the local solution. In subdomains where high gradients are encountered, the grid points are distributed according to geometric progressions, while in subdomains characterized by a smooth solution, a uniform coarse grid is used. For this study, the Burgers and Reynolds equations are employed as representative boundary layer problems in fluid dynamics. The results show that all the finite difference schemes considered, both classical and compact, exhibit qualitatively the same rates of convergence. Moreover, the accuracies achieved by the compact schemes are vastly superior. The high-order schemes require larger nonuniform grid ratios, and hence more grid points to resolve the oscillations. However, as the numerically generated oscillations at the interface between the uniform and nonuniform subdomains are eliminated, compact methods are shown to be superior to second-order and fourth-order mixed methods in accuracy, convergence, and computational efficiency for problems with stiff boundary or interior layers.