Numerical Relativity in Spherical Polar Coordinates: Calculations with the BSSN Formulation Pedro Montero Max-Planck Institute for Astrophysics Garching (Germany) 28/01/13 in collaboration with T.Baumgarte, I. Cordero-Carrion and E. Mueller [arxiv:1211.6632v1; to appear in Phys. Rev. D]
Outline - Astrophysical motivation - Spherical polar coordinates and numerical relativity - BSSN covariant formulation and treatment of singular terms - Partially implicit Runge-Kutta method (PIRK) - Numerical examples: Weak gravitational waves Rotating relativistic stars Schwarzschild black hole - Conclusions 2
Most 3-dimensional numerical relativity codes use Cartesian coordinates. While Cartesian coordinates have many desirable properties, there are astrophysical problems for which spherical polar coordinates are better suited: Accretion onto BHs Collapse of massive stars (SN 1987A) An alternative approach is the use of multi-patch applications 3
Implementing a numerical relativity code in spherical polar coordinates possess several challenges: 1) The original version of the BSSN formulation explicitly assumes Cartesian coordinates. This has been resolved by Brown (2009) who introduced a covariant formulation of the BSSN equations. This formulation is well suited for curvilinear coordinate systems 2) Another challenge is introduced by the coordinate singularities at the origin and at the axis in spherical polar coordinates. 4
Coordinate singularities at the origin and at the axis introduce singular terms in the equations which is a source of numerical problems. One problem arises because of the presence of terms that behave like 1/r in the equations near the origin at r=0 Analytically, regularity of the data (metric) ensures that these terms cancel exactly but on the numerical level is a source of numerical instabilities. Similar problem appears near the axis with terms like 1/sin(θ) 5
Methods to handle these problems 1) Specific gauge choice (i.e. polar/aereal gauges) [Bardeen & Piran 1983; Choptuik 1991]. Disadvantage: restricts the gauge freedom; which is one of the main ingredients in successful evolutions in numerical relativity 2) Regularization method by imposing appropiate parity regularity conditions and local flatness. [Acubierre et al. 2005, Ruiz et al. 2007, Alcubierre et al. 2011] Disadvantage: these are not easy to implement (introduce auxiliary variables and new evolution equations) 6
3) Partially Implicit Runge-Kutta methods: Proposed by Cordero-Carrion et al.(2012) for the solution of the hyperbolic part in the Fully Constrained Formalism of Einstein eqs. PM & Cordero-Carrion,(2012) applied successfully the PIRK method to the BSSN eqs. in spherical coordinates under the assumption of spherical symmetry without the need for regularization at the origin r=0. Terms behaving like 1/r close to r=0 can be numerically interpreted as stiff terms. Advantages: - keep the gauge freedom (use 1+log, Gamma-driver) - No need for a regularization scheme at origin or axis - Simple implementation 7
Implementing the BSSN equations Shibata&Nakamura 1995, Baumgarte&Shapiro 1998, and Brown 2009 (covariant form) Singular terms (e.g. like 1/r) 8
We assume the background metric to be the flat metric is spherical polar coordinates r,θ,φ Conformal metric We treat singular terms analytically by scaling out appropriate powers of r and sinθ Rescaled connection vector We write all evolution equations in terms of h ij, λ i etc... 9
We compute derivatives of the spatial metric as follows: Which is written in terms of h ij We compute derivatives of h ij numerically while all r and sinθ terms are treated analytically 10
2 nd order PIRK method Consider a system of PDEs We assume the L 1 and L 3 differential operators contain only regular terms, whereas L 2 contains the singular terms (i) u is evolved explicitly (ii) v is evolved taking into account the updated value of u for the evaluation of the L 2 operator 11
Computational cost is comparable to explicit methods (no need for analytical or numerical inversion) Source terms in the PIRK operators for the extrinsic curvature evolution equation: 12
Numerical grid Cell-centered grid covering the region 0<r<r max 0<θ<π and 0<ϕ<2π 13
Numerical implementation - Fourth-order finite difference approximation for spatial derivatives. - Second-order PIRK method to evolve in time the hyperbolic equations. - Kreiss-Oliger dissipation term to avoid high frequency noise. - Cell-centered grid to avoid that the location of the puncture at the origin coincides with a grid point in simulations involving a BH. - Outgoing wave boundary conditions at the outer boundary.. 14
I. Weak gravitational waves: axisymmetric waves Small amplitude waves on a flat Minkowski background (l=2,m=0) Results for a numerical grid with (40N,10N,2) with N=1,2 and 4. Outer boundary at r=8.0 1+log gauge condition for the lapse and vanishing shift. h rr for N=4 simulation at θ=1.61 and ϕ=4.71 Crosses: numerical result Solid line: analytical solution L2-norm of the error rescaled by a factor N 4 15
II. Weak gravitational waves: nonaxisymmetric waves Small amplitude waves on a flat Minkowski background (l=2,m=2) Results for a numerical grid with (40,32,64) and outer boundary at r=4.0 1+log gauge condition for the lapse and vanishing shift. h rr for N=4 simulation at θ=1.62 and ϕ=3.19 Crosses: numerical result Solid line: analytical solution Convergence rate is close to 4 th order 16
III. Hydro-without-hydro: rotating relativistic stars Stable relativistic star (Γ=2) rotating at 92% of the allowed massshedding limit. (M~0.85M max and r p /r eq ~0.7) Results for a numerical grid with (48,32,2) and outer boundary at r=25m 1+log gauge condition for the lapse and Gamma-driver condition for the shift. Snapshots of the conformal exponent and the lapse at The initial time and after two spin periods along the pole and the equator. Both profiles remain very similar to the initial data. 17
IV. Schwarzschild trumpet initial data Time independent slicing of Schwarzschild spacetime that satisfies our 1+log slicing condition. Numerical grid of size (160N,2,2) with N=1,2,4 and 8, with outer boundary at r=16m. 2 nd order convergence 18
V. Schwarzschild wormhole initial data Conformal factor: Pre-collapsed lapse: Numerical grid of size (10240,2,2) with outer boundary at r=256m. Coordinate transition from wormhole initial data to timeindependent trumpet data. We plot conformal exponent, lapse and radial orthonormal component of the shift as a function of the gauge-invariant areal radius R 19
V. Schwarzschild wormhole initial data Maximum of the radial shift for different grid sizes (1280N,2,2) for N=1,2,4 and 8 Profiles of the violations of the Hamiltonian constraint at time t=79m. Results are rescaled with N 2. 20
Conclusions - Presented a new numerical relativity code that solves the BSSN equations in spherical polar coordinates without any symmetry assumption. -A key ingredient is the PIRK scheme to integrate the evolution equations in time which allows us to avoid the need for a regularization at the origin or the axis - Obtained the expected stability and convergence of the code - Current developments: AH finder, hydrodynamics, GW extraction and microphysics. 21