Abstract
Explicit Runge-Kutta methods can efficiently be used in the numerical integration of initial value problems for non-stiff systems of ordinary differential equations (ODEs). Let m and p be the number of stages and the order of a given explicit Runge-Kutta method. We have proved in a previous paper [8] that the combination of any explicit Runge-Kutta method with \(m=p\) and the Richardson Extrapolation leads always to a considerable improvement of the absolute stability properties. We have shown in [7] (talk presented at the NM&A14 conference in Borovets, Bulgaria, August 2014) that the absolute stability regions can be further increased when \(p<m\) is assumed. For two particular cases, \(p=3 \wedge m=4\) and \(p=4 \wedge m=6\) it is demonstrated that
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(a)
the absolute stability regions of the new methods are larger than those of the corresponding explicit Runge-Kutta methods with \(p=m\), and
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(b)
these regions are becoming much bigger when the Richardson extrapolation is additionally applied.
The explicit Runge-Kutta methods, which have optimal absolute stability regions, form two large classes of numerical algorithms (each member of any of these classes having the same absolute stability region as all the others). Rather complicated order conditions have to be derived and used in the efforts to obtain some special methods within each of the two classes.
We selected two particular methods within these two classes and tested them by using appropriate numerical examples.
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Acknowledgments
This research is supported in part by Grants DFNI I-01/5 and DFNI I-02/20 from the Bulgarian National Science Found. The authors thanks Center of Scientific Computing at Technical University of Denmark for giving access to their computers for making computations.
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Zlatev, Z., Georgiev, K., Dimov, I. (2015). Selecting Explicit Runge-Kutta Methods with Improved Stability Properties. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2015. Lecture Notes in Computer Science(), vol 9374. Springer, Cham. https://doi.org/10.1007/978-3-319-26520-9_46
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DOI: https://doi.org/10.1007/978-3-319-26520-9_46
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