The 3D wave equation and its Cartesian coordinate stretched perfectly matched embedding – A time-domain Green’s function performance analysis

A general class of 3D perfectly matched, i.e., reflectionless, Cartesian embeddings (perfectly matched layers in the three coordinate directions) is analyzed with the aid of a combined time-domain Green’s function technique and a time-domain, causality-preserving, Cartesian coordinate stretching procedure. It is shown that, for an unbounded embedding of the specified class, the wavefield is, in any 3-rectangular computational solution domain, reproduced exactly. The spurious reflection caused by a (computationally necessary) truncation of the embedding is analyzed as a function of layer thicknesses and their coordinate stretching relaxation functions. A time-domain uniqueness proof for the solution to the truncated embedding problem is provided and a numerical illustration is given for a test case with known analytical solution. For such cases, the pure space-time discretization errors can be separated from the disturbance caused by the spurious reflection. For the second-order coordinate stretched wave equation an equivalent system of first-order equations is presented.