Solving wave equation with spectral methods and nonreflecting boundary conditions

A multidomain spectral method for solving wave equations is presented. This method relies on the expansion of functions on basis of spherical harmonics (Y m l (θ, φ)) for the angular dependence and of Chebyshev polynomials Tn(x) for the radial part. The spherical domains consist of shells surrounding a nucleus and cover the space up to a finite radius R at which boundary conditions are imposed. Time derivatives are estimated using standard finite-differences second order schemes, which are chosen to be implicit to allow for (almost) any size of time-step. Emphasis is put on the implementation of absorbing boundary conditions that allow for the numerical boundary to be completely transparent to the physical wave. This is done using a multipolar expansion of an exact boundary condition for outgoing waves, which is truncated at some point. Using an auxiliary function, which is solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary condition is simply written as a perturbation to the usual Sommerfeld radiation boundary condition. Numerical tests of the method show that very good accuracy can be achieved and and that the quadrupolar part of a wave can pass the numerical boundary without being reflected. This is of particular importance for the simulation of gravitational waves in General Relativity.