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Monotonicity and boundedness in implicit Runge-Kutta methods. (English) Zbl 0583.65045

This paper concerns the analysis of implicit Runge-Kutta methods for approximating the solutions to stiff initial value problems. The analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equation \(U'=\lambda U\) (with \(\lambda\) a complex constant). The properties of monotonicity and boundedness of a method refer to specific moderate rates of growth of the approximations during the numerical calculations. This paper provides necessary conditions for these properties by using the important concept of algebraic stability (introduced by Burrage, Butcher and Crouzeix). These properties will also be related to the concept of contractivity (B- stability) and to a weakened version of contractivity.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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References:

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