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The energy method can be used to identify well-posed initial boundary valueproblems for quasi-linear, symmetric hyperbolic partial differential equationswith maximally dissipative boundary conditions. A similar analysis of thediscrete system can be used to construct stable finite difference equations forthese problems at the linear level. In this paper we apply these techniques tosome test problems commonly used in numerical relativity and observe that whilewe obtain convergent schemes, fast growing modes, or ``artificialinstabilities,'' contaminate the solution. We find that these growing modes canpartially arise from the lack of a Leibnitz rule for discrete derivatives anddiscuss ways to limit this spurious growth.Comment: 18 pages, 22 figure

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