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

  • Authors:

  • Adrianus T. de Hoop;Robert F. Remis;Peter M. van den Berg

  • Affiliations:

  • Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.;Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.;Laboratory of Electromagnetic Research, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands.

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

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.