On the numerical integration of an explicit solution of the homologous collapse’s radial evolution in time

Abstract

ABSTRACT In a recent paper, Slepian and Philcox derive an explicit solution of the homologous collapse from rest of a uniform density sphere under its self-gravity as a function of time. Their solution is given in terms of two curvilinear integrals along a suitable Jordan contour; in practice, it must be approximated by a quadrature rule. The aim of this paper is to examine how the choice of the contour and the quadrature rule affects the accuracy and the efficiency of this integral solution approximation. More precisely, after a study of the complex roots of a transcendental equation that relates time with the variable, some alternative Jordan contours that turn out to be more convenient are proposed. Then, by using as quadrature rule the composite trapezoidal rule because of its reliability and spectral convergence accuracy, some numerical experiments are presented to show that the combination of contours and quadrature rule allows us to obtain numerical results with high accuracy and low computational cost.