Uploaded August 2020 | Updated September 2026, 59 minutes ago
Discontinuous collocation methods and gravitational self-force applications
Michael F O'Boyle,Charalampos Markakis,Pablo D Brubeck,Leor Barack
We present a new approach to solving linear partial differential
equations that occur in calculations of the self-force acting on point particles in orbit around 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. In an effort to exploit such information, we show that high-order accuracy can be recovered by simply adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. Discretizations developed for smooth problems are thus readily extensible to nonsmooth problems. Furthermore, in the context of one-dimensional finite-difference or pseudospectral discretizations, numerical integration and differentiation amount to matrix multiplication. We construct the matrices for such operations, in the presence of known discontinuities, 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.
Discontinuous collocation methods and gravitational self-force applications
Michael F O'Boyle,Charalampos Markakis,Pablo D Brubeck,Leor Barack
We present a new approach to solving linear partial differential
equations that occur in calculations of the self-force acting on point particles in orbit around 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. In an effort to exploit such information, we show that high-order accuracy can be recovered by simply adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. Discretizations developed for smooth problems are thus readily extensible to nonsmooth problems. Furthermore, in the context of one-dimensional finite-difference or pseudospectral discretizations, numerical integration and differentiation amount to matrix multiplication. We construct the matrices for such operations, in the presence of known discontinuities, 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.
