On a numerical algorithm for the solution of the radial Loewner equation

Abstract

The Loewner partial differential equation provides a one-parametric family of conformal maps on the unit disk. The images describe a flow of an expanding simply-connected domain, called the Loewner flow, on the complex plane. In this paper, we present a numerical algorithm for solving the radial Loewner partial differential equation. The algorithm is applied to visualization of Loewner flows with the performance and precision. From the theoretical point of view, our algorithm is based on a recursive formula for determining coefficients of polynomial approximations. We prove that each coefficient converges to true values with reasonable regularity. KEYWORDSconformal mapping, Loewner differential equation, polynomial approximation INTRODUCTIONLoewner theory originates in the work of Loewner in 1923. 1 It provides a one-parametric representation of conformal maps f t on the unit disk D ∶= {z ∈ C ∶ |z| < 1}. The Loewner differential equation, a PDE such a map satisfies, has been successfully used for various problems in Geometric Function Theory, in particular the extremal problems exemplified by the famous Bieberbach conjecture. 2 Recently, it has made remarkable advances in various fields including the cerebrated Schramm-Loewner Evolution (SLE), a stochastic version of the Loewner equation. 3 Among them, a numerical solution of the SLE, 4,5 and the chordal Loewner equation 6 are recently investigated.Nowadays, several variants of families of Loewner-type conformal maps are considered according to the canonical domains and normalizations. Here, we formulate one of the standard models called the radial case, following Pommerenke's characterization. 7,8 ABy the normalization of f ′ t (0), it follows a strict inclusion of the images f s (D) ⊊ f t (D) for all t > s ≥ 0. One of the key properties of a Loewner chain is that it is differentiable with respect to t almost everywhere in t ∈ [0, ∞) and independently in z. Further, it satisfies the Loewner partial differential equation (or Loewner-PDE in short)for all z ∈ D and almost all t ∈ [0, ∞), where the term p(z, t) is called a Herglotz function; measurable with respect to t ∈ [0, ∞) for all fixed z ∈ D, holomorphic with respect to z ∈ D and p(0, t) = 1 for almost all fixed t ∈ [0, ∞) and satisfies 714