Stable higher order finite-difference schemes for stellar pulsation calculations

¹ Institut d’Astrophysique et Géophysique de l’Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium
e-mail: daniel.reese@ulg.ac.be
² Observatoire de Paris, LESIA, CNRS UMR 8109, 92195 Meudon, France

Received: 17 April 2013
Accepted: 5 June 2013

Abstract

Context. Calculating stellar pulsations requires high enough accuracy to match the quality of the observations. Many current pulsation codes apply a second-order finite-difference scheme, combined with Richardson extrapolation to reach fourth-order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist that make fourth-order schemes desirable. A robust and simple finite-difference scheme that can easily be implemented in either 1D or 2D stellar pulsation codes, is therefore required.

Aims. One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grid approach; however, these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions.

Methods. We propose here a new method, inspired from the staggered grid strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order.

Results. This new approach is successfully applied to stellar pulsation calculations, and is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh drift.

Conclusions. Although specifically designed for stellar pulsation calculations, this method can easily be applied to many other physical or mathematical problems.