Abstract
In this paper, we develop an efficient Petrov-Galerkin method for the generalized airfoil equation. In general, the Petrov-Galerkin method for this equation leads to a linear system with a dense coefficient matrix. When the order of the coefficient matrix is large, the complexity for solving the corresponding linear system is huge. For this purpose, we propose a matrix truncation strategy to compress the dense coefficient matrix into a sparse matrix. Subsequently, we use a numerical integration method to generate the fully discrete truncated linear system. At last we solve the corresponding linear system by the multilevel augmentation method. An optimal order of the approximate solution is preserved. The computational complexity for generating the fully discrete truncated linear system and solving it is estimated to be linear up to a logarithm. The spectral condition number of the truncated matrix is proved to be bounded. Numerical examples complete the paper.
Similar content being viewed by others
References
Bland, S.R.: The two-dimensional oscillating airfoil in a wind tunnel in subsonic flow. SIAM. J. Numer. Anal. 18, 830–848 (1970)
Berthold, D., Hoppe, W., Silbermann, B.: A fast algorithm for solving the generalized airfoil equation. J. Comput. Appl. Math. 43, 185–219 (1992)
Capobianco, M.R., Mastroianni, G.: Uniform boundedness of Lagrange operator in some weighted Sobolev-type space. Math. Nachr. 187, 61–77 (1997)
Chen, Z., Wu, B., Xu, Y.: Multilevel augmentation methods for solving operator equations. Numer. Math. J. Chinese Univ. 14, 244–251 (2005)
Golberg, M.A.: Numerical Solution of Integral Equations. Plenum Press, New York (1990)
Hartmann, T., Stephan, E.P.: Rates of convergence for collocation with Jacobi polynomials for the airfoil equation. J. Comput. Appl. Math. 51, 179–191 (1994)
Monegato, G., Sloan, I.H.: Numerical solution of the generalized airfoil equation for an airfoil with a flap. SIAM. J. Numer. Anal. 34, 2288–2305 (1997)
Monegato, G., Lepora, P.: On the numerical resolution of the generalized airfoil equation with Possio kernels. Numer. Math. 56, 775–787 (1990)
Scuderi, L.: A collocation method for the generalized airfoil equation for an airfoil with a flap. SIAM. J. Numer. Anal. 35, 1725–1739 (1998)
Vainikko, G.: Fast solvers of generalized airfoil equations of index 1. Oper. Theory Adv. Appl. 121, 498–516 (2001)
Vainikko, G.: Fast solvers of generalized airfoil equations of index −1. Proc. Est. Acad. Sci. Phys. Math. 50, 145–154 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Doctoral Foundation of Shandong Finance Institute.
Rights and permissions
About this article
Cite this article
Cai, H. An Efficient Algorithm for Solving the Generalized Airfoil Equation. J Sci Comput 37, 251–267 (2008). https://doi.org/10.1007/s10915-008-9241-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-008-9241-5
Keywords
- The generalized airfoil equation
- A matrix truncation strategy
- An optimal approximate order
- The numerical integration method
- The multilevel augmentation method