- Volume 12 Issue 1
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Stability of Explicit Symplectic Partitioned Runge-Kutta Methods
- Koto, Toshiyuki (Department of Information Systems and Mathematical Sciences, Nanzan University) ;
- Song, Eunjee (Department of Information Systems and Mathematical Sciences, Nanzan University)
- Received : 2013.10.09
- Accepted : 2013.12.02
- Published : 2014.03.31
A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic methods for long-term integration. Although it is also related to long-term integration, numerical stability of symplectic methods have received little attention. In this paper, we consider explicit symplectic methods defined for Hamiltonian equations with Hamiltonians of the special form, and study their numerical stability using the harmonic oscillator as a test equation. We propose a new stability criterion and clarify the stability of some existing methods that are visually based on the criterion. We also derive a new method that is better than the existing methods with respect to a Courant-Friedrichs-Lewy condition for hyperbolic equations; this new method is tested through a numerical experiment with a nonlinear wave equation.
- K. Feng, "On the difference schemes and symplectic geometry," in Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, Beijing, China, pp. 42-58, 1985.
- E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed. Heidelberg, Germany: Springer-Verlag, 1993.
- B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics. Cambridge, MA: Cambridge University Press, 2004.
- J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems. London: Chapman & Hall, 1994.
- E. Hairer, "Backward analysis of numerical integrators and symplectic methods," Annals of Numerical Mathematics, vol. 1, pp. 107-132, 1994.
- H. Yoshida, "Recent progress in the theory and application of symplectic integrators," Celestial Mechanics and Dynamical Astronomy, vol. 56, no. 1-2, pp. 27-4, 1993. https://doi.org/10.1007/BF00699717
- F. Y. Liu, X. Wu, and B. K. Lu, "On the numerical stability of some symplectic integrators," Chinese Astronomy and Astrophysics, vol. 31, no. 2, pp. 172-186, 2007. https://doi.org/10.1016/j.chinastron.2007.04.006
- M. A. Lopez-Marcos, J. M. Sanz-Serna, and R. D. Skeel, "An explicit symplectic integrator with maximal stability interval," in Numerical Analysis: A. R. Mitchell 75th Birthday Volume, Singapore: World Scientific, pp.163-175, 1996.
- D. Murai and T. Koto, "Stability and convergence of staggered Runge-Kutta schemes for semilinear wave equations," Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4251-4264, 2011. https://doi.org/10.1016/j.cam.2011.03.020
- G. B. Whitham, Linear and Nonlinear Wave. New York, NY: John Wiley & Sons, 1974.