Abstract
Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl 2-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the “method of lines”.
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Schmitt, B.A. Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations. BIT 28, 884–897 (1988). https://doi.org/10.1007/BF01954908
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DOI: https://doi.org/10.1007/BF01954908