Abstract
In this paper we conduct a detailed numerical analysis of the Gauss-type Nested Implicit Runge-Kutta formulas of order 4, introduced by Kulikov and Shindin in [4]. These methods possess many important practical properties such as high order, good stability, symmetry and so on. They are also conjugate to a symplectic method of order 6 at least. All of these make them efficient for solving many nonstiff and stiff ordinary differential equations (including Hamiltonian and reversible systems). On the other hand, Nested Implicit Runge-Kutta formulas have only explicit internal stages, in the sense that they are easily reduced to a single equation of the same dimension as the source problem. This means that such Runge-Kutta schemes admit a cheap implementation in practice. Here, we check the above-mentioned properties numerically. Different strategies of error estimation are also examined with the purpose of finding an effective one.
This work was supported in part by the National Research Foundation of South Africa under grant No. FA2004033000016.
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Kulikov, G.Y., Shindin, S.K. (2007). Numerical Tests with Gauss-Type Nested Implicit Runge-Kutta Formulas. In: Shi, Y., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds) Computational Science – ICCS 2007. ICCS 2007. Lecture Notes in Computer Science, vol 4487. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72584-8_18
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DOI: https://doi.org/10.1007/978-3-540-72584-8_18
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