Previous |  Up |  Next


energy-preserving; explicit Runge–Kutta methods; gradient
In this paper, Runge-Kutta methods are discussed for numerical solutions of conservative systems. For the energy of conservative systems being as close to the initial energy as possible, a modified version of explicit Runge-Kutta methods is presented. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in energy-preserving to the standard one. Comparing the modified Runge-Kutta method with the standard Runge-Kutta method, numerical experiments are provided to illustrate the effectiveness of the modified Runge-Kutta method.
[1] Brugnano, L., Calvo, M., Montijano, J. I., Rândez, L.: Energy-preserving methods for Poisson systems. J. Comput. Appl. Math. 236 (2012), 3890-3904. DOI 10.1016/ | MR 2926249 | Zbl 1247.65092
[2] Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonian boundary value methods (energy-preserving discrete line integral methods). J. Numer. Anal. Ind. Appl. Math. 5 (2010), 1-2, 17-37. MR 2833606
[3] Calvo, M., Laburta, M. P., Montijano, J. I., Rândez, L.: Error growth in numerical integration of periodic orbits. Math. Comput. Simul. 81 (2011), 2646-2661. DOI 10.1016/j.matcom.2011.05.007 | MR 2822275
[4] Calvo, M., Iserles, A., Zanna, A.: Numerical solution of isospectral flows. Math. Comput. 66 (1997), 1461-1486. DOI 10.1090/S0025-5718-97-00902-2 | MR 1434938 | Zbl 0907.65067
[5] Cooper, G. J.: Stability of Runge-Kutta methods for trajectory problems. IMA J. Numer. Anal. 7 (1987), 1-13. DOI 10.1093/imanum/7.1.1 | MR 0967831 | Zbl 0624.65057
[6] Buono, N. Del, Mastroserio, C.: Explicit methods based on a class of four stage Runge-Kutta methods for preserving quadratic laws. J. Comput. Appl. Math. 140 (2002), 231-243. DOI 10.1016/S0377-0427(01)00398-3 | MR 1934441
[7] Griffiths, D. F., Higham, D. J.: Numerical Methods for Ordinary Differential Equations. Springer-Verlag, London 2010. MR 2759806 | Zbl 1209.65070
[8] Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer-Verlag, Berlin 2002. MR 1904823 | Zbl 1228.65237
[9] Khalil, H. K.: Nonlinear Systems. Third Edition. Prentice Hall, Upper Saddle River, NJ 2002.
[10] Lee, T., Leok, M., McClamroch, N. H.: Lie variational integrators for the full body problem in orbital methanics. Celest. Meth. Dyn. Astr. 98 (2007), 121-144. DOI 10.1007/s10569-007-9073-x | MR 2321987
[11] Li, S.: Introduction to Classical Mechanics. (In Chinese.). University of Science and Technology of China, Hefei 2007.
[12] Shampine, L. F.: Conservation laws and numerical solution of ODEs. Comput. Math. Appl. 12B (1986), 1287-1296. DOI 10.1016/0898-1221(86)90253-1 | MR 0871366
Partner of
EuDML logo