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Frequency evaluation in exponential fitting multistep algorithms for ODEs. (English) Zbl 0996.65075

Summary: We consider the linear multistep algorithms for first-order ordinary differential equations (ODEs) and examine the problem of how the \(\lambda\)-frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms. We find out that the key of the answer consists in analyzing the behaviour of the error. On further investigating the simple case of two-step backwards difference algorithms we produce formulae for the optimal \(\lambda\)’s and show that, if the optimal \(\lambda\)’s are used, the order of the method is increased by one unit. The reported numerical illustrations suggest that further investigations along these lines deserve a real attention.

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
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