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A.R.H. ABS:. The class of one-dimensional stretching functions used in finite-difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function ENTER: NORE 4BI w The class of one-dimensional stretching functions used in finitedifference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions: One isan interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. It was first employed by Thomas et al. The other type of function is a two-sided stretching function, for which the arbitrary slopes at the two ends of the-one-dimensional interval are specified. The -simplest such general function is found to be one based on the inverse tangent." The special case where the ~lopes were both equal and greater than one \'/as first employed by Roberts. The general two-sided function has many applications in the construction of finite-difference grids. An example Qf such an application is found in one of the references.Finite-difference calculations of fluid flow problems are best carried out using an equispaced grid in a rectangular (or cubic) computational domain, with the flow variables and components of the position vector as dependent variables, and boundary conditions applied at the edges (or faces) of the domain. In order to minimize the number of grid points required for a given accuracy, one seeks boundary-fitted coordinate transformations that cluster points in regions where the dependent variables undergo rapid variation. These regions may be due to body geometry (very large curvatures or corners), compressibility (entropy layers, shock waves and contact discontinuities), and viscosity (boundary layers and shear layers). A complex flow may thus contain a variety of suc~ regions of various length scales, and often of unknown location. An ideal grid would-adjust with each time or iteration step to maintain optimum clustering. Such adaptive grid methods, which involve the solution of auxiliary equations, have been developed for one-dimensional problems (refs. 1-3). Their extension to complex multi-dimensional flows ii a difficult problem, particularly when the re~ions requiring clustering do not have simple topological properties required by a finite-difference grid.There are many practical problems in which the locations and length scales of re~ions of rapid variation can be estimated a priori (e.g., known geometry, at...

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