Numerical treatment of the hyperboloidal initial value problem for the vacuum Einstein equations. II. The evolution equations

Abstract

This is the second in a series of articles on the numerical solution of Friedrich's conformal field equations for Einstein's theory of gravity. We will discuss in this paper the numerical methods used to solve the system of evolution equations obtained from the conformal field equations. In particular we discuss in detail the choice of gauge source functions and the treatment of the boundaries. Of particular importance is the process of ''radiation extraction'' which can be performed in a straightforward way in the present formalism.

Numerical Relativity with the Conformal Field Equations

“…The whole system (11,12,13,14,17,18) is then regular in the sense that this point does not have to be located at I + . The vacuum Einstein equationsR ab = 0 then yield…”

“…Note that this equation is not manifestly regular at Ω = 0, but it is actually possible to show that if (11) is satisfied at one point, then by virtue of the other equations (12,13,14,17,18) to be given below, it has to be satisfied everywhere. The whole system (11,12,13,14,17,18) is then regular in the sense that this point does not have to be located at I + .…”

“…5, and has been tested and proven accurate for several types of spacetimes [22]. Frauendiener describes his implementation and some results in [19]. There are two essential problems in these implementations: First of all, the gauge conditions will not usually result in a slicing of I + by cuts of constant Bondi time u.…”

“…When dealing with the physically interesting case of a I + of spherical topology, at least two patches have to be used to represent the Bondi tetrad (l a , n a , m a ,m a ). Frauendiener has achieved to control the movement of I + through the grid by the gauge choice for his formulation [19], in particular the shift vector can be chosen such that I does not change its coordinate location.…”

“…In the original formulation [24,25] and its descendants [26,5,18] Ω (and derivatives) are evolved as dependent variables. All existent numerical codes are based on equations of this type.…”