Dynamics of gravitational waves in 3D: Formulations, methods, and tests

The dynamics of gravitational waves is investigated in full (3+1)-dimensional numerical relativity, emphasizing the difficulties that one might encounter in numerical evolutions, particularly those arising from nonlinearities and gauge degrees of freedom. Using gravitational waves with amplitudes low enough that one has a good understanding of the physics involved, but large enough to enable nonlinear effects to emerge, we study the coupling between numerical errors, coordinate effects, and the nonlinearities of the theory. We discuss the various strategies used in identifying specific features of the evolution. We show the importance of the flexibility of being able to use different numerical schemes, different slicing conditions, different formulations of the Einstein equations [standard Arnowitt, Deser, and Misner vs first order hyperbolic], and different sets of equations (linearized vs full Einstein equations). A nonlinear scalar field equation is presented which captures some properties of the full Einstein equations, and has been useful in our understanding of the coupling between finite differencing errors and nonlinearities. We present a set of monitoring devices which have been crucial in our studying of the waves, including Riemann invariants, pseudo-energy-momentum tensor, Hamiltonian constraint violation, and Fourier spectrum analysis.