Blending curves for landing problems by numerical differential equations, III. Separation techniques

Abstract

A landing curve of an airplane is a smooth curve described by three functions x(s), y(s), and z(s) which are governed by a system of linear ordinary differential equations (ODEs) with certain boundary conditions. The separation techniques are proposed in this paper first for the landing curves of the airplane to the airport. The separation techniques are particularly significant for nonlinear boundary conditions. When a space ship is approaching a shuttle station, the landing curve is ending at the surface of the station. Since the station surface is curved, the ending boundary conditions are nonlinear. This nontrivial nonlinear landing problem can be easily solved by the separation techniques not only for great reduction of computation complexity but also for facile nonlinear error analysis. The error analysis in this paper is made to derive three significant results. (1) The exact blending curves by the separation techniques with the fundamental solutions. (2) The errors O( h 1/2) and O( h 5/2) in H 2 norms of the blending curves by the separation techniques with the cubic Hermite solutions and their a posteriori interpolant, respectively. (3) The errors O( h 2.5) in H 2 norms by those with the fifth-order Hermite method for general cases. Numerical experiments for nonlinear boundary conditions are provided in this paper to display the efficiency of the separation techniques, and to verify the error analysis made.