Stabilization method for coupled channels solutions

Antonio Moro has contributed a a stabilization algorithm to solve or, at least, palliate, the common problem of loss of linear dependence of the coupled channels solutions. Following the suggestion of some Japanese researchers, he implemented a method of re-orthogonalization similar to that described in the papers of Baylis et al, but using a QR orthogonalization procedure instead of the Gram-Schmidt method used by them.

Thanks to this implementation, we are now able to solve efficiently problems which were otherwise not feasible, for example, CDCC calculations with high-lying breakup states, including closed-channels. His group to extend this to the case of transfer reactions, but only with limited success, so in practice it works only for pure coupled channels equations.

A test case dNi-cdcc-QR.in is included in the test/ folder, corresponding to a d+$^{58}$Ni reaction at 20 MeV, including closed channels. In this case, the standard Numerov method fails for JT$>$3, whereas with the stabilization procedure the calculation runs smoothly for all required partial waves.

To activate this option we added a variable hort to the &Fresco namelist, which corresponds to the radial interval step at which stabilization is performed. Typically, try hort = 5 fm. This orthogonalizing continues out to the largest classical turning radius for any open channel, but can continued further by setting the radius variable rmort also in the FRESCO namelist.

WE. Baylis and S.J. Peel, Computer Physics Communications 25 (1982), 7-19,