Numerical integration of Einstein’s field equations

Abstract

Many numerical codes now under development to solve Einstein's equations of general relativity in 3+1 dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three "connection functions". The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability.

General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: the X-ECHO code

“…The 3 + 1 formalism is nowadays adopted in basically all numerical schemes for general relativity (Alcubierre 2008;Baumgarte & Shapiro 2010), where the system of Einstein equations is treated like a Cauchy problem with some initial data to be evolved in time through hyperbolic equations. However, as for the solenoidal condition for the magnetic field, non-evolutionary constraints must be preserved in the numerical evolution, and computational methods for modern codes are divided into two main classes: 1) free-evolution schemes, mainly based on hyperbolic equations alone, where this problem is alleviated by appropriate reformulations of the equations (BSSN: Shibata & Nakamura 1995;Baumgarte & Shapiro 1999), eventually with the addition of propagating modes and damping terms (Z4: Bona et al 2003;Bernuzzi & Hilditch 2010); 2) fully constrained schemes, where the constraints are enforced at each timestep through the solution of elliptic equations (Bonazzola et al 2004), a more robust but computationally demanding option, since elliptic solvers are notoriously difficult to parallelize. Most of the state-of-the-art 3D codes for GRMHD in dynamical spacetimes are based on freeevolution schemes in Cartesian coordinates (Duez et al 2005;Shibata & Sekiguchi 2005;Anderson et al 2006;Giacomazzo & Rezzolla 2007;Montero et al 2008;Farris et al 2008), and have been used for gravitational collapse in the presence of magnetized plasmas Shibata et al 2006a,b;Stephens et al 2007Stephens et al , 2008, evolution of NSs (Duez et al 2006b;Liebling et al 2010), binary NS mergers (Anderson et al 2008;Liu et al 2008;Giacomazzo et al 2009Giacomazzo et al , 2011, and accreting tori around Kerr BHs (Montero et al 2010).…”

Potential for ill-posedness in several second-order formulations of the Einstein equations

“…In the case of vanishing shift vector, the formulation of the Einstein equations referred to as BSSN [5] consists of the following evolution equationṡ…”

“…The case of the standard ADM equations [4] is studied in Section II. The case of the widely used BSSN equations [5] is dealt with in Section III. Finally, the case of the generalizeded Einstein-Christoffel equations (EC) [6], which includes the case of the EinsteinChristoffel equations themselves [7], is developed in Section IV.…”

Bowen-York-type initial data for binaries with neutron stars

“…The code is based on the BSSN formulation of the Einstein equations [32] and the moving puncture gauge condition [33,34]. Maya is very similar to the Einstein code in the Einstein Toolkit [35].…”