Single step methods for general second order singular initial value problems with spherical symmetry (Q1117672)

Consider the class of singular initial value problems of the form (1) \(u''+(\alpha /r)u'=f(r,u,u')\), \(u(r=0)=A\), \(u'(r=0)=0\) with A a finite constant and suitable conditions on f as to guarantee existence and uniqueness of the solution of equation (1). The values \(\alpha =0,1,2\) correspond to plane, cylindrical and spherical geometries, respectively. In the case that f is independent of \(u'\) the application of classical explicit Runge-Kutta-schemes for solving (1) has been studied by several people in the last 25 years. However, those schemes are not self-starting and require some special procedure to obtain the solution at the first step because of the singularity at \(r=0.\) The author obtains a one parameter family of four-stage self-starting explicit Runge-Kutta-Nyström methods in the case \(\alpha =2\). This method is exact for \(u(r)=1/r,1,r,r^ 2,r^ 3,r^ 4\) and has a truncation error of \(0(h^ 5)\) in u and of \(0(h^ 4)\) in \(u'\). If in (1) f is independent of \(u'\), the method has truncation error of \(0(h^ 6)\) in both u and \(u'\).