The constraints as evolution equations for numerical relativity

Abstract

Abstract. The Einstein equations have proven surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be mathematically conserved by the evolution equations. We describe a new approach to rewriting the constraints as first-order evolution equations, thereby guaranteeing that they are satisfied to a chosen accuracy by any discretization scheme. This introduces a set of four subsidiary constraints which are far simpler than the standard constraint equations, and which should be more easily conserved in computational applications. We explore the manner in which the momentum constraints are already incorporated in several existing formulations of the Einstein equations, and demonstrate the ease with which our new constraintconserving approach can be incorporated into these schemes.

Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates

“…Solving the above equations implies that the four constraints are fulfilled by the solution. As already mentioned in the Introduction, some authors have proposed very recently a scheme in which the constraints, rewritten as time evolution equations, are satisfied up to the time discretization errors [20]. On the contrary, in our scheme the constraints are fulfilled within the precision of the space discretization errors (which can be very low with a modest computational cost, thanks to spectral methods).…”

“…requiring a modest CPU time) numerical techniques (based on spectral methods) are now available to solve elliptic equations [18,19]. Very recently some scheme has been proposed in which the constraints, re-written as time evolution equations, are satisfied up to the time discretization error [20]. On the contrary, our scheme guarantees that the constraints are fulfilled within the precision of the space discretization error (which can have a much better accuracy, thanks to the use of spectral methods).…”

Hamiltonian relaxation

“…A method originally developed by Shibata and Nakamura [7] and later used by Baumgarte and Shapiro [8] (BSSN) is today the most commonly used for three-dimensional simulations. The literature offers an ever-growing list of new evolution formulations which can be divided in two groups: unconstrained [3,5,7,8,9,10,11,12,13,14,15,16,17,18,19] and constrained [20,21]. Some of these have been tested numerically under conditions that are either easy to implement numerically and / or have a high degree of symmetry.…”

From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity

“…However, this does not mean that all the resulting evolution systems are necessarily hyperbolic. A clear example is provided by constrained schemes that, due to the elliptic nature of the constraints, lead to mixed elliptic-hyperbolic systems -see however [274]. Among the many possible criteria, in the present general discussion we will classify 3+1 evolution systems according to the presence or absence of elliptic equations in the system.…”