Generalized Spline Interpolation of Functions with Large Gradients in Boundary Layers

Abstract

The spline interpolation of functions of one variable with large gradients in boundary layers is studied. It is well known that applying polynomial splines to interpolate functions of this kind leads to significant errors when the small parameter is comparable with the grid step size. A generalized spline that is an analogue of a cubic spline is constructed. The spline is exact on the component responsible for large gradients of the function in the boundary layer. The boundary-layer component is considered as a function of a general form; in particular, the case of an exponential boundary layer is treated. The existence, uniqueness, and accuracy of the constructed spline are analyzed.