New Stability Results for Explicit Runge-Kutta Methods
Provides theoretical bounds for numerical analysts designing stable Runge-Kutta methods, though the results are incremental extensions of known techniques.
The paper derives sharp bounds on the absolute and parabolic stability radii of explicit Runge-Kutta methods using polynomial theory, including Walsh's theorem and Bernstein bases. It also provides inequalities relating stability radii across methods of different orders and stages.
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.