Hyperbolic Formulations and Numerical Relativity

We present our numerical comparisons between weakly hyperbolic, strongly hyperbolic, and symmetric hyperbolic systems based on Ashtekar's connection variables. Our numerical code demonstrates gravitational wave propagation in plane symmetric space-time, and we compare the stability and/or accuracy by monitoring the violation of the constraints. We also study alternative approaches to obtain stable evolutions, that can be called "asymptotically constrained systems". We demonstrate "λ-system" and "adjusted-system" for the Ashtekar system, and show they work as desired. We propose a new mechanism to control the stability by evaluating the eigenvalues of constraint propagation equations, and this mechanism affects more than the system's symmetric hyperbolicity.