Necessary and Sufficient Conditions for Strong Stability of Explicit Runge–Kutta Methods

Strong stability is a property of time integration schemes for ODEs that preserve temporal monotonicity of solutions in arbitrary (inner product) norms. It is proved that explicit Runge–Kutta schemes of order \(p\in 4{\mathbb {N}}\) with \(s=p\) stages for linear autonomous ODE systems are not strongly stable, closing an open stability question from [Z. Sun and C.-W. Shu, SIAM J. Numer. Anal. 57 (2019), 1158–1182]. Furthermore, for explicit Runge–Kutta methods of order \(p\in {\mathbb {N}}\) and \(s>p\) stages, we prove several sufficient as well as necessary conditions for a less restrictive notion of strong stability. These conditions involve both the stability function and the hypocoercivity index of the ODE system matrix. This index is a structural property combining the Hermitian and skew-Hermitian part of the system matrix.