A numerical method and efficient preconditioner for generalized airfoil equations

In this paper, the generalized airfoil equations with Cauchy algebraic singularity are researched by introducing the collocation method to conquer the singularity of the integral equation. Theoretical analysis and examples showed this method has accuracy O(hm+1)O(hm+1) with m the order of the interpolate polynomial. In addition, by separating the singular part into two so that they can offset against each other in the collocation points, a new equivalent integrate function is also constructed. Moreover, to accelerate the convergence of solving the obtained linear system, a tridiagonal preconditioner is also presented for the singular integral operator by the operator splitting idea. Compared to the traditional tridiagonal preconditioner, our method is more easily computed by an explicit inverse. Finally, some numerical examples are presented to illustrate the behavior of the collocation method and this preconditioner.