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Acoustical and Canonical Fluid Dynamics in numerical general relativity

Periodic Reporting for period 2 - ACFD (Acoustical and Canonical Fluid Dynamics in numerical general relativity)

Reporting period: 2018-12-15 to 2019-12-14

What is the problem/issue being addressed?

Binary neutron stars are astrophysical scale particle colliders, where all four fundamental forces come into play. Gravitational and electromagnetic waves, emitted during their inspiral and merger, carry valuable information about weak and strong nuclear interactions in their interior, where matter reaches the most extreme densities known in nature. Detection and parameter estimation of gravitational wave signals requires precise modelling of their sources. Numerical relativity and computational fluid dynamics are vital for simulating the inspiral of these systems and computing their gravitational waves.

Why is it important for society?

The first observed event, GW170817, had striking implications. It resolved the 50-year mystery of the origin of short gamma-ray bursts; it provided strong evidence for mergers as the main source of half the heaviest elements (the r-process elements); and gave an independent measurement of the Hubble constant. Future events, from galaxies not too far away, can also address a major goal of multimessenger astrophysics: From the imprint of tides on inspiral waveforms, we can measure the radius and tidal deformation of the inspiraling stars and infer the behavior of cold matter above nuclear density.

What are the overall objectives?

We develop a new, well-posed formulation of relativistic fluid dynamics, based on Hamilton's principle. We utilize this canonical formulation of the Euler-Einstein system in order to perform high-precision computational fluid dynamic simulations in numerical general relativity. The programme contributes towards a deeper understanding of fluid dynamics in curved spacetime, and how Kelvin's circulation theorem, Helmholtz's third theorem, and other the relevant features imprint themselves in gravitational waveforms. It explores the utility of such waveforms as probes of dense matter physics.
We have developed a formulation of relativistic hydrodynamics consisting of two exact conservation laws without source terms.
The relativistic Euler equation has been replaced by a simpler, Hamilton-Jacobi equation for the velocity potential, or its gradient, a covariant flux-conservative form of Hamilton’s equation.
We have performed a hyperbolicity analysis which proves that the formulation is well-posed for positive sound speed.
We have demonstrated that Kelvin's circulation theorem and its corollary, Helmholtz's third theorem, hold exactly during binary neutron star inspiral.
We developed a constraint-damping scheme that preserves these theorems during simulations in numerical general relativity.
We have performed a variety of numerical tests, including Bondi accretion, propagation of smooth or discontinuous perturbations, and neutron star oscillations.
Our work explores a fruitful synergy between numerical relativity, hydrodynamics and gravitational wave science. The system has novel features, that are not shared by standard schemes derived from stress-energy conservation. It goes beyond the state of the art, as it allows exact conservation of circulation in 2D and 3D numerical simulations. This has impact on numerical general relativity, gravitational wave astrophysics, and computational fluid dynamics in general. We have also discovered that the scheme exhibits supercovnergent behavior (cf. figure) for flows that are close to stationary, such as an accreting flow onto a static black hole. By the end of the project, we intend to explore if this superconvergence can be attained in simulations of binary neutron stars on circular orbits, which are quasi-stationary in a rotating frame, and possess an approximate constant of motion: the Hamiltonian in the rotating frame (the analogue of the Jacobi constant in the restricted circular three body problem) is constant throuhout the star. We will also explore a method of circumventing the conservative to primitive conversion required by conventional relativistic hydrodynamic schemes, and can take up to ~40% of computational time or lead to numerical errors that often terminate expensive supercomputer simulations.
Bondi accretion onto a Schwarzschild black hole: superconvergence of Hamiltonian fluid dynamics