NRPyElliptic: A Fast Hyperbolic Relaxation Elliptic Solver for Numerical Relativity, I: Conformally Flat, Binary Puncture Initial Data

TL;DR

NRPyElliptic introduces a hyperbolic relaxation elliptic solver within the NRPy+ framework for numerical relativity, significantly speeding up initial data setup for binary black holes compared to traditional methods.

Contribution

It presents a novel hyperbolic relaxation scheme for elliptic PDEs in numerical relativity, making the solver easier to modify and faster than existing elliptic solvers.

Findings

Accelerates relaxation waves, achieving many orders of magnitude speedup.

Requires only 0.3% of the runtime of full BBH simulations compared to Einstein Toolkit.

Still about 12 times slower than TwoPunctures for initial data setup.

Abstract

We introduce NRPyElliptic, an elliptic solver for numerical relativity (NR) built within the NRPy+ framework. As its first application, NRPyElliptic sets up conformally flat, binary black hole (BBH) puncture initial data (ID) on a single numerical domain, similar to the widely used TwoPunctures code. Unlike TwoPunctures, NRPyElliptic employs a hyperbolic relaxation scheme, whereby arbitrary elliptic PDEs are trivially transformed into a hyperbolic system of PDEs. As consumers of NR ID generally already possess expertise in solving hyperbolic PDEs, they will generally find NRPyElliptic easier to tweak and extend than other NR elliptic solvers. When evolved forward in (pseudo)time, the hyperbolic system exponentially reaches a steady state that solves the elliptic PDEs. Notably NRPyElliptic accelerates the relaxation waves, which makes it many orders of magnitude faster than the usual constant-wavespeed approach. While it is still ${\sim}12$x slower than TwoPunctures at setting up full-3D BBH ID, NRPyElliptic requires only ${\approx}0.3\%$ of the runtime for a full BBH simulation in the Einstein Toolkit. Future work will focus on improving performance and generating other types of ID, such as binary neutron star.