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from Part IV - Initial Value Problems for Ordinary Differential Equations
Published online by Cambridge University Press: 29 September 2022
The topic of this chapter is linear stability of schemes for ODEs. The notions of stiffness, linear stability domain, and A-stability are introduced. We discuss the A-stability of Runge-Kutta schemes via the amplification factor. The notion of A-acceptable, Pade approximations of the exponential, and the Hairer-Wanner-Norsett theorem are then presented. This allows us to show that all Gauss-Legendre-Runge-Kutta schemes are A-stable. For multistep schemes we present linear stability criteria and Dahlquist second barrier theorem. The boundary locus method concludes the chapter.
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