Discontinuous collocation methods and gravitational self-force applications

Abstract

Numerical simulations of extreme mass ratio inspirals, the most important 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.

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

“…The Hermite integration methods outlined in this paper have also been shown to work when a point-particle source term is added to the flat spacetime Klein-Gordon equation. In this case, the Hermite integration methods must be modified to accommodate discontinuous functions, which we have shown in [34] for the second order method. (The appropriate generalization for the fourth order method will be presented in a subsequent paper).…”

“…(2.1). In particular, discontinuous Hermite rules can be obtained with the method of undetermined coefficients, which can be generalized to accommodate jump discontinuities across distributional sources (see [34] for details). Distributional source terms arise, for instance, in the motion of a particle orbiting a black hole.…”

Symmetric integration of the 1+1 Teukolsky equation on hyperboloidal foliations of Kerr spacetimes

“…Discretization implies that the spatial derivative operators ∂ σ and ∂ 2 σ in Eq. (2.22) will amount to matrices, which can be computed using polynomial collocation (finite-difference or pseudo-spectral) methods [32,36]. We use D ı and D (2) ı to denote the first and second order differentiation matrices respectively:…”

“…But this is beyond the scope of this paper and may not generalize straightforwardly to discon-Chebyschev differentiation matrices have a high condition number which increases as N 2 or N 4 for first and second order derivatives respectively, and matrix multiplications involve summations over elements that are not compensated by typical numerical linear algebra libraries; therefore round-off error exceeds truncation error for a high number of points. tinuous collocation methods [32], which is a natural next step towards including a point particle source. Instead, for these scenarios, we use a finite-difference grid with more points to gain the accuracy needed.…”

“…In earlier work, it was also shown (cf. [32,36]) that time-symmetric integration methods are unconditionally stable and preserve energy and the U(1) gauge charge when used to evolve linear perturbations of Schwarzschild black holes. Using such methods to compute the gravitational self-force or radiation reaction from extreme mass ratio inspiral ensures that energy and angular momentum is lost only to true radiation, rather than numerical error, and yields the correct physical observables.…”