Cauchy-characteristic Evolution of Einstein-Klein-Gordon Systems: The   Black Hole Regime
  The Cauchy+characteristic matching (CCM) problem for the scalar wave equation is investigated in the background geometry of a Schwarzschild black hole. Previously reported work developed the CCM framework for the coupled Einstein-Klein-Gordon system of equations, assuming a regular center of symmetry. Here, the time evolution after the formation of a black hole is pursued, using a CCM formulation of the governing equations perturbed around the Schwarzschild background. An extension of the matching scheme allows for arbitrary matching boundary motion across the coordinate grid. As a proof of concept, the late time behavior of the dynamics of the scalar field is explored. The power-law tails in both the time-like and null infinity limits are verified. 
Philippos Papadopoulos, Pablo Laguna