Numerical methods for analytic continuation and mesh generation

The problem of computing the analytic continuation of a holomorphic function known on a circle is considered. Several fast numerical schemes based on solving an initial-value problem for the Cauchy-Riemann equations are analyzed. To avoid instability problems, some of the schemes consist of two parts: one for integrating the Cauchy-Riemann equations, and one for smoothing the function values so obtained. We show that with appropriate integration and smoothing methods, the stability and accuracy of such schemes is sufficient for many applications. The schemes are well-suited for generating level curves and stream lines of conformal mappings. Computed examples are presented. We also indicate how the schemes can be used to generate near-orthogonal boundary-fitted grids with given mesh sizes along the boundary.