Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I would like to get some insight into the practicalities of applying conformal mapping techniques for the numerical solution of PDEs. Up until now I had the impression that conformal mapping techniques basically enable the solution of specific PDEs on complex domains by transforming the $\mathbf{solution}$ on a simpler one via the conformal map. Is this true?

Assuming I have Laplace's equation in 2D I can solve it based on a cartesian grid using finite differences. In case the computational domain is more complicated, lets say a half annulus the solution to the equation should be obtained by "warping" the obtained solution with the conformal map. However if we consider the same problem and use a solution method based on a curvilinear coordinate system fitted to the half annulus and the finite difference method then the finite difference formulas that have to be applied change considerably yielding a different solution.

What I was wondering about is if the initial assumption is false? I have heard about the Joukowski transform being applied for the computation of streamlines around airfoils using the transformed analytical solution around a cylinder, that's the reason I had this assumption.

share|improve this question
add comment

1 Answer

I would put in a slightly different way: "Conformal mapping enables the solution of specific boundary value problems by transforming the BVP to a simpler domain via a conformal map, and then transporting the solution back."

Using the chain rule, you can check that $\Delta (u\circ f) = (\Delta(u)\circ f) |f'|^2$ whenever $f$ is conformal and $u$ is sufficiently smooth. In particular, $\Delta u\equiv 0$ if and only if $\Delta (u\circ f)\equiv 0$. Thus, instead of solving Laplace's equation on the original (complicated) domain, we may decide to solve it on a disk or half-plane instead. Of course, we must keep track of what happens to the boundary conditions. And we must be able to compute a conformal map, e.g., by the Schwarz-Christoffel method.

To get the general idea of what is going on, I recommend this illustrated explanation by Tobin Driscoll.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.