Following previous work in vacuum spacetimes, we investigate theconstraint-damping properties in the presence of matter of the recentlydeveloped traceless, conformal and covariant Z4 (CCZ4) formulation of theEinstein equations. First, we evolve an isolated neutron star with an ideal gasequation of state and subject to a constraint-violating perturbation. Wecompare the evolution of the constraints using the CCZ4 andBaumgarte-Shibata-Shapiro-Nakamura-Oohara-Kojima (BSSNOK) systems. Second, westudy the collapse of an unstable spherical star to a black hole. Finally, weevolve binary neutron star systems over several orbits until the merger, theformation of a black hole, and up to the ringdown. We show that the CCZ4formulation is stable in the presence of matter and that the constraintviolations are one or more orders of magnitude smaller than for the BSSNOKformulation. Furthermore, by comparing the CCZ4 and the BSSNOK formulationsalso for neutron star binaries with large initial constraint violations, weinvestigate their influence on the errors on physical quantities. We also givea new, simple and robust prescription for the damping parameter that removesthe instabilities found when using the fully covariant version of CCZ4 in theevolution of black holes. Overall, we find that at essentially the samecomputational costs the CCZ4 formulation provides solutions that are stable andwith a considerably smaller violation of the Hamiltonian constraint than theBSSNOK formulation. We also find that the performance of the CCZ4 formulationis very similar to another conformal and traceless, but noncovariantformulation of the Z4 system, i.e. the Z4c formulation.
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