Commun. Comput. Phys., 12 (2012), pp. 1275-1292.

Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

Qin Sheng 1*, Hai-Wei Sun 2

1 Center for Astrophysics, Space Physics and Engineering Research, Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA.
2 Department of Mathematics, University of Macau, Macao.

Received 10 August 2011; Accepted (in revised version) 9 January 2012
Available online 3 May 2012


This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number. Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense. Simulated examples are given to illustrate the conclusion.

AMS subject classifications: 65M06, 65M12, 65F35, 15A12

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Key words: Paraxial equation, highly oscillatory problems, eikonal splitting, asymptotic stability, matrix eigenvalues, spectral radius.

*Corresponding author.
Email: (Q. Sheng), (H. Sun)

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