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Parallel one-step methods with minimal parallel stages. (English) Zbl 0840.65079

Suppose that sufficiently many processors are available, so that embedded methods do not need to be considered. Then, for certain types of method, lower bounds for the number of (parallel) stages, \(s_p\), required to achieve a given order, \(p\), are derived. For the general explicit one step method it is shown that \(s_p \geq p\). Results of this type are obtained for implicit Runge-Kutta methods, (singly) diagonally implicit Runge-Kutta methods, semi-implicit Runge-Kutta methods and Rosenbrock-Wanner methods.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65Y05 Parallel numerical computation
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