Effective computation of optimal stability polynomials

Abstract

The construction of stable explicit multistage Runge-Kutta methods in 1950–1960 stumbled over a certain extremal problem for polynomials. The solution to this problem is known as the optimal stability polynomial and its computation is notoriously difficult. We propose a new method for the effective evaluation of optimal stability polynomials which is based on the explicit analytical representation of the solution. The main feature of the method is its independence of the computational complexity of the degree of the solution.