Computer Science – Numerical Analysis
Scientific paper
Jan 2012
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2012hell.confr..23g&link_type=abstract
10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011. Edited by Io
Computer Science
Numerical Analysis
Scientific paper
This is a review of the so-called "complex-plane strategy" (abbreviated 'CPS') and its application to astrophysical problems. CPS is an efficient alternative for problems obeying differential equations with terms that become undefined on the real axis when the independent variable r becomes greater than a particular value R (the radius, say). If so, then the decisive alternative proposed by CPS is to integrate the differential equations of the problem in the complex plane and, particularly, on a complex path. Such a complex path can be, for instance, a straight line parallel to the real axis and at a small imaginary distance from it. One may wonder why to continue numerical integrations beyond R; the reason is simply that, in this case, we can compute accurately all the dependent functions of the problem over a sufficiently extended interval of the independent variable r (up to r=2R, say). Then, to compute critical physical and geometrical characteristics of the model under consideration, we can interpolate in such extended function tables, instead of extrapolating in the othewise reduced function tables (terminating at r=R). In numerical analysis, interpolation is a safe and accurate procedure; while extrapolation suffers from large errors and, in most cases, becomes unreliable. CPS has been applied to astrophysical problems, in which the well-known polytropic equation of state is involved, or other equations of state with similar mathematical characteristics. Such problems are: the classical stellar polytropic models, the white dwarf models obeying Chandrasekhar's equation of state, the solar and the jovian systems, and the general-relativistic polytropic models simulating neutron stars. Finally, a new software package for accurately integrating in the complex plane along general complex paths is described and proposed, and some comparisons with existing packages are presented.
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