Numerische Mathematik

, Volume 36, Issue 4, pp 405–429

Solving Theodorsen's integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods

  • Martin H. Gutknecht

DOI: 10.1007/BF01395955

Cite this article as:
Gutknecht, M.H. Numer. Math. (1981) 36: 405. doi:10.1007/BF01395955


We investigate several iterative methods for the numerical solution of Theodorsen's integral equation, the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors. All our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast Fourier transform for this. In particular, we discuss the nonlinear JOR iteration, the nonlinear SOR iteration, a nonlinear second order Euler iteration, the nonlinear Chebyshev semi-iterative method, and its cyclic variant. Under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters (e.g., the underrelaxation factor) and the asymptotic convergence factor. Weaker related results for the general non-symmetric case are presented too. Practically, our methods extend the range of application of Theodorsen's method and improve its effectiveness strikingly.

Subject Classifications

AMS (1980): 30C30 42A50, 65H10, 65R20 CR: 5.19 5.15, 5.18 

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Martin H. Gutknecht
    • 1
  1. 1.Seminar für angewandte MathematikETH-Zentrum HGZürichSwitzerland