Numerical stability of the Z4c formulation of general relativity

We study numerical stability of different approaches to the discretization of a conformal decomposition of the Z4 formulation of general relativity. We demonstrate that in the linear, constant coefficient regime a novel discretization for tensors is formally numerically stable with a method of lines time integrator. We then perform a full set of apples with apples'' tests on the nonlinear system, and thus present numerical evidence that both the new and standard discretizations are, in some sense, numerically stable in the nonlinear regime. The results of the Z4c numerical tests are compared with those of Baumgarte-Shapiro-Shibata-Nakamura-Oohara-Kojima (BSSNOK) evolutions. We typically do not employ the Z4c constraint damping scheme and find that in the robust stability and gauge wave tests the Z4c evolutions result in lower constraint violation at the same resolution as the BSSNOK evolutions. In the gauge wave tests, we find that the Z4c evolutions maintain the desired convergence factor over many more light-crossing times than the BSSNOK tests. The difference in the remaining tests is marginal.