The discrete energy method in numerical relativity: towards long-term stability

LUIS LEHNER, DAVID NEILSEN, OSCAR REULA, MANUEL TIGLIO

CLASSICAL AND QUANTUM GRAVITY

Año: 2004 vol. 21 p. 5819 - 5848

0264-9381

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete system can be used to construct stable finite difference equations for these problems at the linear level. In this paper we apply these techniques to some test problems commonly used in numerical relativity and observe that while we obtain convergent schemes, fast growing modes, or ``artificial instabilities,'' contaminate the solution. We find that these growing modes can partially arise from the lack of a Leibnitz rule for discrete derivatives and discuss ways to limit this spurious growth.