Hyperelliptic thetafunctions and spectral methods
Abstract
A code for the numerical evaluation of hyperelliptic thetafunctions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for solutions of the Ernst equation where the branch points of the Riemann surface are parameterized by the physical coordinates. An exploration of the whole parameter space of the solution is thus only possible with an efficient code. The use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision. The case of almost degenerate Riemann surfaces is addressed. Tests of the numerics using identities for periods on the Riemann surface and integral identities for the Ernst potential and its derivatives are performed. It is shown that an accuracy of the order of machine precision can be achieved. These accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary Einstein equations. The resulting solution agrees with the thetafunctional solution to very high precision.
 Publication:

Journal of Computational and Applied Mathematics
 Pub Date:
 May 2004
 arXiv:
 arXiv:nlin/0512065
 Bibcode:
 2004JCoAM.167..193F
 Keywords:

 Hyperelliptic thetafunctions;
 Spectral methods;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematics  Algebraic Geometry
 EPrint:
 25 pages, 12 figures