Achieving Brouwer’s law with implicit Runge–Kutta methods
- 115 Downloads
In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times.
Key wordsround-off error probabilistic error propagation implicit Runge–Kutta methods long-time integration efficient implementation
Unable to display preview. Download preview PDF.
- 2.K. R. Grazier, W. I. Newman, J. M. Hyman, P. W. Sharp, and D. J. Goldstein, Achieving Brouwer’s law with high-order Störmer multistep methods, ANZIAM J., 46 (2004/05), pp. C786–C804.Google Scholar
- 3.E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn., Springer Ser. Comput. Math., vol. 31, Springer, Berlin, 2006.Google Scholar
- 4.P. Henrici, The propagation of round-off error in the numerical solution of initial value problems involving ordinary differential equations of the second order, in Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-Differential Equations (Rome 1960), Birkhäuser, Basel, 1960, pp. 275–291.Google Scholar