Necessary and sufficient conditions for strong stability of explicit Runge-Kutta methods
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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∈ 4ℕ 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∈ℕ and s>p stages, we prove several sufficient as well as necessary conditions for 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.
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