Difference Schemes Based on Exponentially Converging Quadratures for the Cauchy Integral

Conventional finite difference schemes are based on interpolating a grid function with a polynomial of finite degree. The error of such schemes decreases as a certain power of the step size. In this work, a fundamentally new class of finite difference schemes with exponential convergence rate is proposed, which is radically faster than the traditional polynomial rate. A typical gain in accuracy reaches 5–8 orders of magnitude or more. The proposed approach is uniformly applicable to various classes of problems in mathematical physics and is demonstrated using boundary value problems for ordinary differential equations (ODEs) as an example. Examples are provided to illustrate the advantages of the proposed approach.