SENR/NRPy+: Numerical Relativity in Singular Curvilinear Coordinate Systems

TL;DR

SENR/NRPy+ is an open-source numerical relativity code that efficiently models black hole dynamics in singular curvilinear coordinates, demonstrating exponential convergence and ease of use with Python interfaces.

Contribution

It extends BSSN formulation to broader curvilinear coordinates, providing a Python-based tensorial equation interface and validated high-accuracy black hole simulations.

Findings

Achieves nearly exponential convergence of errors with increasing finite difference order.

Demonstrates high accuracy in black hole evolutions with coordinate singularities.

Validates against established codes with excellent agreement.

Abstract

We report on a new open-source, user-friendly numerical relativity code package called SENR/NRPy+. Our code extends previous implementations of the BSSN reference-metric formulation to a much broader class of curvilinear coordinate systems, making it ideally-suited to modeling physical configurations with approximate or exact symmetries. In the context of modeling black hole dynamics, it is orders of magnitude more efficient than other widely used open-source numerical relativity codes. NRPy+ provides a Python-based interface in which equations are written in natural tensorial form and output at arbitrary finite difference order as highly efficient C code, putting complex tensorial equations at the scientist's fingertips without the need for an expensive software license. SENR provides the algorithmic framework that combines the C codes generated by NRPy+ into a functioning numerical relativity code. We validate against two other established, state-of-the-art codes, and achieve excellent agreement. For the first time we demonstrate--in the context of puncture, trumpet, and dual black hole evolutions--nearly exponential convergence of constraint violation and gravitational waveform errors to zero as the order of spatial finite difference derivatives is increased, while holding the spherical-like coordinate grids fixed at moderate resolution. Such behavior outside the horizons is remarkable, as numerical errors do not converge to zero inside horizons, and all points along the polar axis are coordinate singularities. The formulation addresses such coordinate singularities via cell-centered grids and a simple change of basis that analytically regularizes tensor components with respect to the coordinates. Future plans include extending this formulation to allow dynamical coordinate grids and bispherical-like distribution of points to efficiently capture orbiting compact binary dynamics.