Compressing variable-coefficient exterior Helmholtz problems via RKFIT

Druskin, Vladimir and Güttel, Stefan and Knizhnerman, Leonid (2016) Compressing variable-coefficient exterior Helmholtz problems via RKFIT. [MIMS Preprint]

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Abstract

The efficient discretization of Helmholtz problems on unbounded domains is a challenging task, in particular, when the wave medium is nonhomogeneous. We present a new numerical approach for compressing finite difference discretizations of such problems, thereby giving rise to efficient perfectly matched layers (PMLs) for nonhomogeneous media. This approach is based on the solution of a nonlinear rational least squares problem using the RKFIT method proposed in [M. Berljafa and S. G���¼ttel, SIAM J. Matrix Anal. Appl., 36(2):894--916, 2015]. We show how the solution of this least squares problem can be converted into an accurate finite difference grid within a rational Krylov framework. Several numerical experiments are included. They indicate that RKFIT computes PMLs more accurate than previous analytic approaches and even works in regimes where the Dirichlet-to-Neumann functions to be approximated are highly irregular. Spectral adaptation effects allow for accurate finite difference grids with point numbers below the Nyquist limit.

Item Type: MIMS Preprint
Uncontrolled Keywords: finite difference grid, Helmholtz equation, Dirichlet-to-Neumann map, perfectly matched layer, rational approximation, continued fraction
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 30 Functions of a complex variable
MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Stefan Güttel
Date Deposited: 04 Nov 2016
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2511

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