Maximal Radius for Stabilty Regions of Explicit Runge Kutta Methods by Real Quantifier Elimination

Explicit Runge Kutta (ERK) methods are well known for numerical solutions of initial value problems. Since the stability function of any ERK scheme is a polynomial, these methods cannot be A-stable. However, the size and shape of the their stability regions can matter. By careful selection of the parameters of the m-stage method that appear in the coefficients of the stability polynomial, one can influence the shape and size of stability region and can aim to optimize it in some sense. Hong et al. in their seminal paper in 1997 [8] showed how real quantifier elimination (RQE) can be used for the investigation of stability problems of dynamical systems. They also showed some examples which were too difficult for their RQE program called QEPCAD. Following their setting and Seip et al. work [15], we investigate the maximal largest possible disk and half-disk that can be inscribed in the stability region of an ERK method and for a whole class of stability regions associated to the class of m-stage pth order methods by RQE. The main tool for the investigation is Mathematica’s RQE implementation. We obtain exact optimal radii for the family of m-stage pth-order methods, when p = m - 1 or p = m-2 (m ≤ 5). We also show that the maximal radius for the particular Dormand-Prince 7-stage 5th-order method can be computed in a few seconds.