Splines under tension in integral transfer problems

Abstract

Splines under tension are used to develop an interpolatory representation of integral radiation operators. A quantitative measure of the stability of this representation is derived and approximated in terms of local grid lengths and the tension parameter. Cubic splines (zero tension parameter) are shown to be approximately stable when the ratio of successive grid lengths lies within a specific range. Scaling procedures for determining the tension constant from the structure of the grids and the characteristics of the problem are discussed, and the utility of these procedures is illustrated by application to the formation of resonance lines in two-dimensional media with exponential height variation of opacity.