This is the second paper in a series describing a numerical implementation of the conformal Einstein equation. This paper deals with the technical details of the numerical code used to perform numerical time evolutions from a "minimal" set of data. We outline the numerical construction of a complete set of data for our equations from a minimal set of data. The second and the fourth order discretisations, which are used for the construction of the complete data set and for the numerical integration of the time evolution equations, are described and their efficiencies are compared. By using the fourth order scheme we reduce our computer resource requirements -with respect to memory as well as computation time -by at least two orders of magnitude as compared to the second order scheme.
This is the first of a series of papers describing a numerical implementation of the conformally rescaled Einstein equation, an implementation designed to calculate asymptotically flat spacetimes, especially spacetimes containing black holes.In the present paper we derive the new first order time evolution equations to be used in the scheme. These time evolution equations can either be written in symmetric hyperbolic or in flux-conservative form. Since the conformally rescaled Einstein equation, also called the conformal field equations, formally allow us to place the grid boundaries outside the physical spacetime, we can modify the equations near the grid boundaries and get a consistent and stable discretisation. Even if we calculate spacetimes containing black holes, there is no need for introducing artifical boundaries in the physical spacetime, which then would complicate, influence, or even exclude the computation of certain spacetime regions.
We use the conformal approach to numerical relativity to evolve
hyperboloidal gravitational wave data without any symmetry assumptions.
Although our grid is finite in space and time, we cover the whole future of
the initial data in our calculation, including future null and future
timelike infinity.
A formalism and its numerical implementation is presented which allows one to calculate quantities determining the spacetime structure in the large directly. This is achieved by conformal techniques by which null infinity (ti) and future timelike infinity (i+) are mapped to grid points on the numerical grid. The determination of the causal structure of singularities, the localization of event horizons, the extraction of radiation, and the avoidance of unphysical reflections at the outer boundarv of the erid. are demonstrated with calculations of spherically symmetric models with a --I scalar field as matter and radiation model.PACS number(s): 04.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.