Theodorsen integral equation

Theodorsen's integral equation [a7] is a well-known tool for computing numerically the conformal mapping $g$ of the unit disc $D$ onto a star-like region $\Delta$ given by the polar coordinates $\tau$, $\rho ( \tau )$ of its boundary $\Gamma$. The mapping $g$ is assumed to be normalized by $g ( 0 ) = 0$, $g ^ { \prime } ( 0 ) > 0$. It is uniquely determined by its boundary correspondence function $\theta$, which is implicitly defined by

\begin{equation*} g ( e ^ { i t } ) = \rho ( \theta ( t ) ) e ^ { i \theta ( t ) } ( \forall t \in \mathbf{R} ), \end{equation*}

\begin{equation*} \int _ { 0 } ^ { 2 \pi } \theta ( t ) d t = 2 \pi ^ { 2 }. \end{equation*}

Theodorsen's equation follows from the fact that the function $h ( w ) : = \operatorname { log } ( g ( w ) / w )$ is analytic in $D$ and can be extended to a homeomorphism of the closure $\overline{ D }$ onto the closure $\overline{\Delta}$. It simply states that the $2 \pi$-periodic function $y$: $t \mapsto \theta - t$ is the conjugate periodic function of $x$: $t \mapsto \operatorname { log } \rho ( \theta ( t ) )$, that is, $y = K x$, where $K$ is the conjugation operator defined on $L [ 0,2 \pi ]$ by the principal value integral

\begin{equation*} ( K x ) ( t ) : = \frac { 1 } { 2 \pi } \text{P} \cdot \text{V} \cdot \int _ { 0 } ^ { 2 \pi } x ( s ) \operatorname { cot } \frac { t - s } { 2 } d s \ (a.e.) . \end{equation*}

When restricted to $L _ { 2 } [ 0,2 \pi ]$, $K$ is a skew-symmetric endomorphism of norm $1$ with a very simple diagonal representation in Fourier space: when $x$ has the real Fourier coefficients $a_0 , a_1 , \dots$, $b _ { 1 } , b _ { 2 } , \dots$, then $y$ has the coefficients $0 , - b _ { 1 } , - b _ { 2 } , \dots$, $a _ { 1 } , a _ { 2 } , \dots$.

Hence, while Theodorsen's integral equation is normally written as

\begin{equation*} \theta ( t ) - t = \frac { 1 } { 2 \pi } \operatorname {P} \cdot \operatorname {V}\cdot \int _ { 0 } ^ { 2 \pi } \operatorname { log } \rho ( \theta ( s ) ) \operatorname { cot } \frac { t - s } { 2 } d s, \end{equation*}

for practical purposes the conjugation is executed by transformation to Fourier space: $x$ is approximated by a trigonometric polynomial of degree $N$, whose Fourier coefficients are quickly found by the fast Fourier transform, which then can also be applied to determine values at $2 N$ equi-spaced points of the trigonometric polynomial that approximates $y = K x$ (cf. also Fourier series). Before the fast Fourier transform became the standard tool for this discrete conjugation process, the transition from the values of $x$ to those of $y$ was based on multiplication by a matrix, called the Wittich matrix in [a1]. The fast Fourier transform meant a cost reduction from $O ( N ^ { 2 } )$ to $O(N \log N)$ operations per iteration.

Until the end of the 1970s the recommendation was to solve a so-obtained discrete version of Theodorsen's equation by fixed-point (Picard) iteration, an approach that is limited to Jordan regions with piecewise differentiable boundary satisfying $| \rho ^ { \prime } / \rho | < 1$, and is very slow when the bound $1$ is nearly attained. Other regions, like those from airfoil design, which was the standard application targeted by T. Theodorsen, could be handled by using first a suitable preliminary conformal mapping, which turned the exterior of the wing cross-section into the exterior of a Jordan curve that is close to a circle; see [a6], Chapt. 10. Moreover, for this application, the equation has to be modified slightly to map the exterior of the disc onto the exterior of a Jordan curve.

M. Gutknecht [a4], [a2] extended the applicability of Theodorsen's equation by applying more refined iterative methods and discretizations, and O. Hübner [a5] improved the convergence order from linear to quadratic by adapting R. Wegmann's treatment of a similar equation obtained by choosing $h ( w ) : = g ( w ) / w$ instead. Wegmann's method [a9], [a10] applies the Newton method and solves the linear equation for the corrections by interpreting it as a Riemann–Hilbert problem that can be solved with four fast Fourier transforms.

A common framework for conformal mapping methods based on function conjugation is given in [a3]; Theodorsen's restriction to regions given in polar coordinates can be lifted. Both Theodorsen's [a8] and Wegmann's [a11] equations and methods can be extended to the doubly connected case.

References

[a1]	D. Gaier, "Konstruktive Methoden der konformen Abbildung" , Springer (1964) Zbl 0132.36702
[a2]	M.H. Gutknecht, "Numerical experiments on solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods" SIAM J. Sci. Statist. Comput. , 4 (1983) pp. 1–30
[a3]	M.H. Gutknecht, "Numerical conformal mapping methods based on function conjugation" J. Comput. Appl. Math. , 14 (1986) pp. 31–77
[a4]	M.H. Gutknecht, "Solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various nonlinear iterative methods" Numer. Math. , 36 (1981) pp. 405–429
[a5]	O. Hübner, "The Newton method for solving the Theodorsen equation" J. Comput. Appl. Math. , 14 (1986) pp. 19–30
[a6]	P.K. Kythe, "Computational conformal mapping" , Birkhäuser (1998)
[a7]	T. Theodorsen, "Theory of wing sections of arbitrary shape" Rept. NACA , 411 (1931)
[a8]	T. Theodorsen, I.E. Garrick, "General potential theory of arbitrary wing sections" Rept. NACA , 452 (1933)
[a9]	R. Wegmann, "Ein Iterationsverfahren zur konformen Abbildung" Numer. Math. , 30 (1978) pp. 453–466
[a10]	R. Wegmann, "An iterative method for conformal mapping" J. Comput. Appl. Math. , 14 (1986) pp. 7–18 (Translated from German) (English translation of [9])
[a11]	R. Wegmann, "An iterative method for the conformal mapping of doubly connected regions" J. Comput. Appl. Math. , 14 (1986) pp. 79–98