Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation

Abstract

The long-time evolution of extreme mass-ratio inspiral systems requires minimal phase and dispersion errors to accurately compute far-field waveforms, while high accuracy is essential near the smaller black hole (modeled as a Dirac delta distribution) for self-force computations. Spectrally accurate methods, such as nodal discontinuous Galerkin (DG) methods, are well suited for these tasks. Their numerical errors typically decrease as $\propto (\Delta x{)}^{N+1}$, where $\Delta x$ is the subdomain size and $N$ is the polynomial degree of the approximation. However, certain DG schemes exhibit superconvergence, where truncation, phase, and dispersion errors can decrease as fast as $\propto (\Delta x{)}^{2N+1}$. Superconvergent numerical solvers are, by construction, extremely efficient and accurate. We theoretically demonstrate that our DG scheme for the scalar Teukolsky equation with a distributional source is superconvergent, and this property is retained when combined with the hyperboloidal layer compactification technique. This ensures that waveforms, total energy and angular-momentum fluxes, and self-force computations benefit from superconvergence. We empirically verify this behavior across a family of hyperboloidal layer compactifications with varying degrees of smoothness. Additionally, we show that self-force quantities for circular orbits, computed at the point particle's location, also exhibit a certain degree of superconvergence. Our results underscore the potential benefits of numerical superconvergence for efficient and accurate gravitational waveform simulations based on DG methods.

14 pages, 7 figures, and 2 tables. Accepted to the General Relativity and Gravitation journal's topical collection on hyperboloidal foliations. Version 2 includes an extended discussion of long-lived "junk tails" at future null infinity and additional dissipative self-force computations

References in corpus (4)

• Intermediate and Extreme Mass-Ratio Inspirals -- Astrophysics, Science Applications and Detection using LISA
• Hyperboloidal foliations and scri-fixing
• Toward exponentially-convergent simulations of extreme-mass-ratio inspirals: A time-domain solver for the scalar Teukolsky equation with singular source terms
• Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution

Cited by in corpus (1)

• Topical Collection-Hyperboloidal Foliations in the Era of Gravitational-Wave Astronomy: From Mathematical Relativity to Astrophysics