Symmetric multistep methods for constrained Hamiltonian systems

详细信息    查看全文

• 作者：Paola Console (1)
  Ernst Hairer (1)
  Christian Lubich (2)
  
• 关键词：65L06 ; 65L80 ; 65P10
• 刊名：Numerische Mathematik
• 出版年：2013
• 出版时间：July 2013
• 年：2013
• 卷：124
• 期：3
• 页码：517-539
• 全文大小：372KB
• 参考文献：1. Andersen, H.C.: Rattle: a 鈥渧elocity鈥?version of the shake algorithm for molecular dynamics calculations. J. Comput. Phys. 52, 24鈥?4 (1983) CrossRef
  2. Ar茅valo, C., F眉hrer, C., S枚derlind, G.: $\beta $-Blocked multistep methods for Euler鈥揕agrange DAEs: linear analysis. Z. Angew. Math. Mech. 77(8), 609鈥?17 (1997) CrossRef
  3. Console, P., Hairer, E.: Long-term stability of symmetric partitioned linear multistep methods. In: Current Challenges in Stability Issues for Numerical Differential Equations, C.I.M.E. Summer Sch., pp. 1鈥?6. Springer, Berlin (2013)
  4. Hairer, E.: Backward error analysis for multistep methods. Numer. Math. 84, 199鈥?32 (1999) CrossRef
  5. Hairer, E., Hairer, M.: GniCodes鈥擬atlab programs for geometric numerical integration. In: Frontiers in Numerical Analysis (Durham 2002), pp. 199鈥?40. Springer, Berlin (2003)
  6. Hairer, E., Lubich, C.: Symmetric multistep methods over long times. Numer. Math. 97, 699鈥?23 (2004) CrossRef
  7. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006)
  8. Jay, L.: Symplectic partitioned Runge鈥揔utta methods for constrained Hamiltonian systems. SIAM J. Numer. Anal. 33, 368鈥?87 (1996) CrossRef
  9. Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18, 189鈥?02 (1976) CrossRef
  10. Quinlan, G.D., Tremaine, S.: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694鈥?700 (1990)
  11. Reich, S.: Symplectic integration of constrained Hamiltonian systems by composition methods. SIAM J. Numer. Anal. 33, 475鈥?91 (1996) CrossRef
  12. Ryckaert, J.-P., Ciccotti, G., Berendsen, H.J.C.: Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of $n$-alkanes. J. Comput. Phys. 23, 327鈥?41 (1977) CrossRef
  
• 作者单位：Paola Console (1)
  Ernst Hairer (1)
  Christian Lubich (2)
  
  1. Dept. de Math茅matiques, Univ. de Gen猫ve, 1211, Gen猫ve 4, Switzerland
  2. Mathematisches Institut, Universit盲t T眉bingen, Auf der Morgenstelle, 72076, T眉bingen, Germany
  
• ISSN：0945-3245

文摘

A method of choice for the long-time integration of constrained Hamiltonian systems is the Rattle algorithm. It is symmetric, symplectic, and nearly preserves the Hamiltonian, but it is only of order two and thus not efficient for high accuracy requirements. In this article we prove that certain symmetric linear multistep methods have the same qualitative behavior and can achieve an arbitrarily high order with a computational cost comparable to that of the Rattle algorithm.