An iterative method for solving nonlinear Riemann-Hilbert problems
The author considers the following nonlinear Riemann-Hilbert boundary value problem (RHP): Given a function F find a function w analytic in the unit disc D such that the boundary values \(w(e^{it})=u(t)+iv(t)\) satisfy the equation \(F(t,u(t),v(t))=0\) for all t. The equation \(F(t,u,v)=0\) defines for each t a curve \(M_ t\). If all these curves are homeomorphic to the unit circle RHP is called of type (A). It is of type (B) if all the \(M_ t\) are homeomorphic to the real line. After reviewing shortly the conditions for the existence and uniqueness of solutions to RHP of either type, the author proposes an iterative method which is based on linearization and takes advantage of the fact that for a linear RHP \(a(t)u(t)+b(t)v(t)=c(t)\) a solution can be constructed explicitly by means of the operator K of conjugation. It is the method which the reviewer has developed for the problem of conformal mapping (which is a RHP of type A). The reviewer's method of proof is carried over and yields quadratic convergence of the iteration. The numerical solution of the linear RHP can be done very efficiently using fast Fourier transform. On a grid of N points the cost per iteration is O(N log N). Two numerical examples are presented. Some peculiarities of the convergence of the numerical procedure are described. These were explained in the meantime by the reviewer [J. Comput. Appl. Math. 29, 207-224 (1990; Zbl 0695.30005)].
nonlinear Riemann-Hilbert boundary value problem
scientific article; zbMATH DE number 4157762
scientific article