Abstract We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High-order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the implicit boundary integral formulation of surface integrals involving the Laplace layer kernels.
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Boundary signature of singularity in the presence of a shock wave
Matter falling into a Schwarzschild-AdS black hole from the left causes increased focussing of ingoing geodesics from the right, and, as a consequence, they reach the singularity sooner. In a standard Penrose diagram, the singularity “bends down”. We show how to detect this feature of the singularity holographically, using a boundary two-point function. We model the matter with a shock wave, and show that this bending down of the singularity can be read off from a novel analytic continuation of the boundary two-point function. Along the way, we obtain a generalization of the recently proposed thermal product formula for two-point correlators.
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- Award ID(s):
- 2107939
- PAR ID:
- 10511676
- Publisher / Repository:
- SciPost Physics
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 16
- Issue:
- 2
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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