Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1D colliding gravitational plane waves

Abstract

We investigate how the accuracy and stability of numerical relativity simulations of 1D colliding plane waves depends on choices of equation formulations, gauge conditions, boundary conditions, and numerical methods, all in the context of a first-order 3+1 approach to the Einstein equations, with basic variables some combination of first derivatives of the spatial metric and components of the extrinsic curvature tensor. Hyperbolic schemes, specifically variations on schemes proposed by Bona and Massó and Anderson and York, are compared with variations of the Arnowitt-Deser-Misner formulation. Modifications of the three basic schemes include raising one index in the metric derivative and extrinsic curvature variables and adding a multiple of the energy constraint to the extrinsic curvature evolution equations. Redundant variables in the Bona-Massó formulation may be reset frequently or allowed to evolve freely. Gauge conditions which simplify the dynamical structure of the system are imposed during each time step, but the lapse and shift are reset periodically to control the evolution of the spacetime slicing and the longitudinal part of the metric. We show that physically correct boundary conditions, satisfying the energy and momentum constraint equations, generically require the presence of some ingoing eigenmodes of the characteristic matrix. Numerical methods are developed for the hyperbolic systems based on decomposing flux differences into linear combinations of eigenvectors of the characteristic matrix. These methods are shown to be second-order accurate, and in practice second-order convergent, for smooth solutions, even when the eigenvectors and eigenvalues of the characteristic matrix are spatially varying.

Scalar glueball, scalar quarkonia, and their mixing

“…With the lowest one corresponding to m π as low as 180 MeV, it is found that, when the quark mass is smaller than that of the strange, a 0 mass levels off, in contrast to those of a 1 and other hadrons that have been calculated on the lattice. This confirms the trend that has been observed in earlier works at higher quark masses in both the quenched and unquenched calculations [7,8,9,10]. The chiral extrapolated mass a 0 = 1.42 ± 0.13 GeV suggests that a 0 (1450) is a qq state.…”

“…However, there are several serious problems with this scenario. First, the isovector scalar meson a 0 (1450) is confirmed to be a qq meson in lattice calculations [6,7,8,9, 10] which will be discussed later. As such, the degeneracy of a 0 (1450) and K * 0 (1430), which has a strange quark, cannot be explained if M S is larger than M N by ∼ 250 MeV.…”

A wave propagation algorithm for hyperbolic systems on curved manifolds

“…These quantities are evaluated atQ, whereQ is the same quantity as computed in step (3). (11) De-orthonormalize at the cell edge:…”

“…If the solution domain is a flat manifold but contains complicated internal or external boundaries, it is often possible to introduce a curvilinear grid that conforms to the boundaries [41]. A fundementally different situation arises when the solution domain is a curved manifold, such as the surface of a sphere of radius r embedded in lR 3 . In this situation, the curvature of the manifold modifies the underlying dynamics.…”

“…The method is numerically conservative, second order accurate in smooth regions, non-dispersive in regions of large gradients, and shockcapturing. This method has been successfully applied in the past to several applications areas, including gas dynamics [21], acoustics [9,10], elasticity and plasticity [8,22,26,27], combustion and detonation waves [16,17], relativistic hydrodynamics [1], and numerical relativity [3,18].…”