Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations

Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like $$h^{p+4}\exp (h^2Lt)$$hp+4exp(h2Lt), where $$p$$p is the order of the method, and $$L$$L depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.

[1]  E. Hairer,et al.  Long-Term Stability of Symmetric Partitioned Linear Multistep Methods , 2014 .

[2]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[3]  The Control of Parasitism in G-symplectic Methods , 2014, SIAM J. Numer. Anal..

[4]  G. Bergeles,et al.  Notes on Numerical Fluid Mechanics and Multidisciplinary Design , 2012 .

[5]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[6]  E. Hairer,et al.  G-symplecticity implies conjugate-symplecticity of the underlying one-step method , 2013 .

[7]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[8]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[9]  E. Hairer,et al.  On conjugate symplecticity of B-series integrators , 2013 .

[10]  J. C. Butcher,et al.  Dealing with Parasitic Behaviour in G-Symplectic Integrators , 2013 .

[11]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[12]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[13]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[14]  Beatrice Paternoster,et al.  Numerical integration of Hamiltonian problems by G-symplectic methods , 2014, Adv. Comput. Math..

[15]  Ernst Hairer,et al.  Symmetric multistep methods over long times , 2004, Numerische Mathematik.

[16]  Numerische,et al.  Backward error analysis for multistep methods , 2022 .

[17]  Desmond J. Higham,et al.  Numerical Analysis 1997 , 1997 .

[18]  Ernst Hairer,et al.  Oscillations over long times in numerical Hamiltonian systems , 2009 .

[19]  Ernst Hairer,et al.  Order barriers for symplectic multi-value methods , 1998 .

[20]  Arieh Iserles,et al.  Highly Oscillatory Problems , 2009 .