arXiv subjects

Michael F. O'Boyle

Publications and source records attributed to Michael F. O'Boyle.

6 recordsLinked to original sources

arXiv2023

Discontinuous collocation and symmetric integration methods for distributionally-sourced hyperboloidal partial differential equations

This work outlines a time-domain numerical integration technique for linear hyperbolic partial differential equations sourced by distributions (Dirac $δ$-functions and their derivatives). Such problems arise when studying binary black hole systems in the extreme mass ratio limit. We demonstrate that such source terms may be converted to effective domain-wide sources when discretized, and we introduce a class of time-steppers that directly account for these discontinuities in time integration. Moreover, our time-steppers are constructed to respect time reversal symmetry, a property that has been connected to conservation of physical quantities like energy and momentum in numerical simulations. To illustrate the utility of our method, we numerically study a distributionally-sourced wave equation that shares many features with the equations governing linear perturbations to black holes sourced by a point mass.

math.NA↗

arXiv2022

Conservative Evolution of Black Hole Perturbations with Time-Symmetric Numerical Methods

The scheduled launch of the LISA Mission in the next decade has called attention to the gravitational self-force problem. Despite an extensive body of theoretical work, long-time numerical computations of gravitational waves from extreme-mass-ratio-inspirals remain challenging. This work proposes a class of numerical evolution schemes suitable to this problem based on Hermite integration. Their most important feature is time-reversal symmetry and unconditional stability, which enables these methods to preserve symplectic structure, energy, momentum and other Noether charges over long time periods. We apply Noether's theorem to the master fields of black hole perturbation theory on a hyperboloidal slice of Schwarzschild spacetime to show that there exist constants of evolution that numerical simulations must preserve. We demonstrate that time-symmetric integration schemes based on a 2-point Taylor expansion (such as Hermite integration) numerically conserve these quantities, unlike schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This makes time-symmetric schemes ideal for long-time EMRI simulations.

gr-qc↗

arXiv2021

Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.

math.NA↗

arXiv2020

A Parametrized Equation of State for Neutron Star Matter with Continuous Sound Speed

We present a generalized piecewise polytropic parameterization for the neutron-star equation of state using an ansatz that imposes continuity in not only pressure and energy density, but also in the speed of sound. The universe of candidate equations of state is shown to admit preferred dividing densities, determined by minimizing an error norm consisting of integral astrophysical observables. Generalized piecewise polytropes accurately reproduce astrophysical observables, such as mass, radius, tidal deformability and mode frequencies, as well as thermodynamic quantities, such as the adiabatic index. This makes the new EOS useful for Bayesian parameter estimation from gravitational waveforms. Moreover, since they are differentiable, generalized piecewise polytropes can improve pointwise convergence in numerical relativity simulations of neutron stars. Existing implementations of piecewise polytropes can easily accommodate this generalization.

astro-ph.HE↗

arXiv2019

Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration

Numerical evolution of time-dependent differential equations via explicit Runge-Kutta or Taylor methods typically fails to preserve symmetries of a system. It is known that there exists no numerical integration method that in general preserves both the energy and the symplectic structure of a Hamiltonian system. One is thus normally forced to make a choice. Nevertheless, a symmetric integration formula, obtained by Lanczos-Dyche via two-point Taylor expansion (or Hermite interpolation), is shown here to preserve both energy as well as symplectic structure for linear systems. This formula shares similarities with the Euler-Maclaurin formula, but is superconvergent rather than asymptotically convergent. For partial differential equations, the resulting evolution methods are unconditionally stable, i.e, not subject to a Courant-Friedrichs-Lewy limit. Although generally implicit, these methods become explicit for linear systems.

math.NA↗

arXiv2018

Joint Spectral Characterization of Photon-Pair Sources

The ability to determine the joint spectral properties of photon pairs produced by the processes of spontaneous parametric downconversion (SPDC) and spontaneous four wave mixing (SFWM) is crucial for guaranteeing the usability of heralded single photons and polarization-entangled pairs for multi-photon protocols. In this paper, we compare six different techniques that yield either a characterization of the joint spectral intensity or of the closely-related purity of heralded single photons. These six techniques include: i) scanning monochromator measurements, ii) a variant of Fourier transform spectroscopy designed to extract the desired information exploiting a resource-optimized technique, iii) dispersive fibre spectroscopy, iv) stimulated-emission-based measurement, v) measurement of the second-order correlation function $g^{(2)}$ for one of the two photons, and vi) two-source Hong-Ou-Mandel interferometry. We discuss the relative performance of these techniques for the specific cases of a SPDC source designed to be factorable and SFWM sources of varying purity, and compare the techniques' relative advantages and disadvantages.

quant-ph↗