Constraint damping in the Z4 formulation and harmonic gauge

We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a lambda-system as suggested by Brodbeck et al (1999 J. Math. Phys. 40 909). We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore, the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.