A spectral collocation method for singularly perturbed hyperbolic equations and its application to a point particle around a Schwarzschild black hole
We consider a hyperbolic conservation law with singular source terms which involve the dirac delta function or its derivative. Preliminary results of a convergence study of the spectral collocation method are presented. For a linear hyperbolic equation with a delta function source term, the spectral collocation methods can yield fast convergence without regularization, due to a novel cancellation of oscillations on collocation grid points. The linear wave equation obtained for a Schwarzschild black hole is solved using the spectral collocation method. Wave forms produced under these conditions are a primary source for gravitational wave detectors such as LISA and LIGO. The result of the spectral collocation methods in this situation is also compared to commonly used finite-difference methods such the Lax-Wendroff scheme.