Researchers from Goethe University Frankfurt and the Frankfurt Institute for Advanced Studies (FIAS) report accurate simulation of merging black holes using code developed by European ExaHyPE project which has been working to develop an Exascale Hyperbolic PDE Engine for numerical simulation code for gravitational waves. The idea is to produce code able to run on the next generation of exascale machines.

Solving Einstein’s equations efficiently and accurately is the challenge. This work, led by Professor Luciano Rezzolla of Goethe, used a novel numerical method that employs the ideas of the Russian physicist Galerkin and allows the computation of gravitational waves on supercomputers with very high accuracy and speed.

“Reaching this result, which has been the goal of many groups worldwide for many years, was not easy,” said Prof. Rezzolla in an announcement of the results on EurekaAlert. “Although what we accomplished is only a small step toward modelling realistic black holes, we expect our approach to become the paradigm of all future calculations.”

It is dense work. Here is the abstract taken from their paper published last month in APS Physical Review D[i]:

“We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system.

The resulting governing partial differential equations system is written in nonconservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.”

The EurekaAlert article noted that while they are waiting for the first “exascale” computers to be built, the ExaHyPE scientists are already testing their software at the largest supercomputing centres available in Germany. “The biggest ones are those at the Leibniz supercomputing centre LRZ in Munich, and the high-performance computing centre HLRS in Stuttgart. These computers are already constructed with more than 100,000 processors and will become much larger soon.”

“The most exciting aspect of the ExaHyPE project is the unique combination of theoretical physics, applied mathematics and computer science,” said Michael Dumbser, leader of the applied mathematics team in Trento. “Only the combination of these three different disciplines allows us to exploit the potential of supercomputers for understanding the complexity of the universe.”

Link to EurekaAlert article: https://www.eurekalert.org/pub_releases/2018-05/guf-bhf052818.php

Link to APS paper: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.97.084053

Link to ExaHyPE: https://exahype.eu

[i]Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes,  https://journals.aps.org/prd/abstract/10.1103/PhysRevD.97.084053