Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915493137350656 |
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| author | Vishal, Manas Field, Scott E. Gottlieb, Sigal Ryan, Jennifer |
| author_facet | Vishal, Manas Field, Scott E. Gottlieb, Sigal Ryan, Jennifer |
| contents | 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 $\propto (Δx)^{N+1}$, where $Δx$ 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 $\propto (Δx)^{2N+1}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation Vishal, Manas Field, Scott E. Gottlieb, Sigal Ryan, Jennifer General Relativity and Quantum Cosmology Numerical Analysis 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 $\propto (Δx)^{N+1}$, where $Δx$ 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 $\propto (Δx)^{2N+1}$. 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. |
| title | Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation |
| topic | General Relativity and Quantum Cosmology Numerical Analysis |
| url | https://arxiv.org/abs/2503.11523 |