# I Applications needing fast numerical conformal maps

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1. Sep 15, 2016

### Diane Wilbor

What are useful practical applications of numerical conformal mapping that are most limited by map computation speed or boundary complexity? I'm betting some of the applications will be be physics PDEs, so I chose this DE subforum to ask.

As part of an engineering project I've implemented several numerical methods including Kantorovich's method of simultaneous equations, as well as Fornberg's iterative version, and Driscoll/Trefethen's classic numerical Schwarz-Christoffel mapping methods. But some experiments with FFT based methods were really successful. These started as variants of Fornberg's FFT method but with some tweaks, modern numerical libraries, and reorganizing the algorithm to eliminate memory bandwidth limits. The performance became surprisingly, almost startlingly, good, especially the memory-aware organization. So I have a working prototype of a numerical conformal mapping code that can compute the mapping of even huge boundaries of a million line or curve segments in fractions of a second. Its performance is beyond my expectations and needs.

So with this serendipidous new experimental tool, I'm trying to find any applications in any field which have been limited by conformal mapping speed and/or boundary complexity.

The number of applications of conformal mapping is huge of course, and a great resource (with hundreds of application references) has been the comprehensive but dated Conformal Mapping book by Schinzinger and Laura. Every single one of the (hundreds) of examples in that book have shapes and domains which are simple enough that they've been easy to quickly map even with Matlab. Most of those 1980's era examples use only 10 or so sides or curves! But today, for example, mapping a 2D airfoil boundary defined by 500 line segments is no challenge to existing tools with just one second of compute. Does anything need 50K? 5M?

But my question is asking about identifying specifically speed or complexity limited applications of numerical conformal mapping. Is there some research project, some engineering application, some important or interesting boundary-limit PDE which would be helped if it could only handle contours of a million edges, and/or solve them in milliseconds?

Thanks so much!
-Diane

2. Sep 21, 2016