The current important issue in numerical relativity is to determine which formulation of the Einstein equations provides us with stable and accurate simulations. Based on our previous work on "asymptotically constrained" systems, we here present constraint propagation equations and their eigenvalues for the ArnowittDeser-Misner (ADM) evolution equations with additional constraint terms (adjusted terms) on the right-hand side. We conjecture that the system is robust against violation of constraints if the amplification factors (eigenvalues of the Fourier component of the constraint propagation equations) are negative or purely imaginary. We show that such a system can be obtained by choosing multipliers of the adjusted terms. Our discussion covers Detweiler's proposal and Frittelli's analysis, and we also mention the so-called conformaltraceless ADM systems.
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