New Stability Results for Explicit Runge-Kutta Methods

TL;DR

This paper uses polynomial polar forms and geometric inequalities to establish sharp bounds on the stability regions of explicit Runge-Kutta methods, enhancing understanding of their stability properties.

Contribution

It introduces new bounds on stability radii for explicit Runge-Kutta methods using advanced polynomial and geometric techniques.

Findings

Derived sharp bounds on the absolute stability radius.

Established bounds on the parabolic stability radius.

Provided inequalities relating stability radii across different methods.

Abstract

The theory of polar forms of polynomials is used to provide for sharp bounds on the radius of the largest possible disc (absolute stability radius), and on the length of the largest possible real interval (parabolic stability radius), to be inscribed in the stability region of an explicit Runge-Kutta method. The bounds on the absolute stability radius are derived as a consequence of Walsh's coincidence theorem, while the bounds on the parabolic stability radius are achieved by using Lubinsky-Ziegler's inequality on the coefficients of polynomials expressed in the Bernstein bases and by appealing to a generalized variation diminishing property of Bezier curves. We also derive inequalities between the absolute stability radii of methods with different orders and number of stages.