Worldtube excision method for intermediate-mass-ratio inspirals: Scalar-field toy model

The computational cost of inspiral and merger simulations for black hole binaries increases in inverse proportion to the square of the mass ratio q := m(2)/m(1) <= 1. One factor of q comes from the number of orbital cycles, which is proportional to 1/q, and another is associated with the required number of time steps per orbit, constrained (via the Courant-Friedrichs-Lewy condition) by the need to resolve the two disparate length scales. This problematic scaling makes simulations progressively less tractable at smaller q. Here we propose and explore a method for alleviating the scale disparity in simulations with mass ratios in the intermediate astrophysical range (10(-4) less than or similar to q less than or similar to 10(-2)), where purely perturbative methods may not be adequate. A region of radius much larger than m(2) around the smaller object is excised from the numerical domain, and replaced with an analytical model approximating a tidally deformed black hole. The analytical model involves certain a priori unknown parameters, associated with unknown bits of physics together with gauge-adjustment terms; these are dynamically determined by matching them to the numerical solution outside the excision region. In this paper we develop the basic idea and apply it to a toy model of a scalar charge in a circular-geodesic orbit around a Schwarzschild black hole, solving for the massless Klein-Gordon field in a eth (1 + 1)D framework. Our main goal here is to explore the utility and properties of different matching strategies, and to this end we develop two independent implementations, a finite-difference one and a spectral one. We discuss the extension of our method to a full 3D numerical evolution and to gravity.