Abstract
In this article, the stability property and the error analysis of higher-order exponential variational integration are examined and discussed. Toward this purpose, at first we recall the derivation of these integrators and then address the eigenvalue problem of the amplification matrix for advantageous choices of the number of intermediate points employed. Obviously, the latter determines the order of the numerical accuracy of the method. Following a linear stability analysis process we show that the methods with at least one intermediate point are unconditionally stable. Finally, we explore the behavior of the energy errors of the presented schemes in prominent numerical examples and point out their excellent efficiency in long term integration.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
B. Engquist, A. Fokas, E. Hairer, A. Iserles, Highly Oscillatory Problems (Cambridge University Press, Cambridge, 2009)
J. Wendlandt, J.E. Marsden, Mechanical integrators derived from a discrete variational principle. Phys. D, 106, 223–246 (1997)
C. Kane, J.E. Marsden, M. Ortiz, Symplectic-energy-momentum preserving variational integrators. J. Math. Phys. 40, 3353–3371 (2001)
J.E. Marsden, M. West, Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)
E. Hairer, C. Lubich, G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method. Acta Numer. 12, 399–450 (2003)
B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics (Cambridge Monographs on Applied and Computational Mathematics, Cambridge, 2004)
S. Ober-Blöbaum, Galerkin variational integrators and modified symplectic Runge–Kutta methods. IMA J. Numer. Anal. 37, 375–406 (2017)
O.T. Kosmas, D.S. Vlachos, Phase-fitted discrete Lagrangian integrators. Comput. Phys. Commun. 181, 562–568 (2010)
O.T. Kosmas, D.S. Leyendecker, Analysis of higher order phase fitted variational integrators. Adv. Comput. Math. 42, 605–619 (2016)
O.T. Kosmas, S. Leyendecker, Variational integrators for orbital problems using frequency estimation. Adv. Comput. Math. 45, 1–21 (2019). https://doi.org/10.1007/s10444-018-9603-y
J. Hersch, Contribution a la methode des equations aux differences. Z. Angew. Math. Phys. 9, 129–180 (1958)
J. Certaine, The Solution of Ordinary Differential Equations with Large Time Constants. Mathematical Methods for Digital Computers (Wiley, New York, 1960), pp. 128–132
D.A. Pope, An exponential method of numerical integration of ordinary differential equations. Commun. ACM 6, 491–493 (1963)
P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30, 177–189 (1979)
M. Hochbruck, C. Lubich, H. Selfhofer, Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)
B. García-Archilla, M.J. Sanz-Serna, R.D. Skeel, Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1999)
A. Nealen, M. Mueller, R. Keiser, E. Boxerman, M. Carlson, Physically based deformable models in computer graphics. Comput. Graph. Forum 25, 809–836 (2006)
O.T. Kosmas, S. Leyendecker, Phase lag analysis of variational integrators using interpolation techniques. PAMM Proc. Appl. Math. Mech. 12, 677–678 (2012)
O.T. Kosmas, D. Papadopoulos, Multisymplectic structure of numerical methods derived using nonstandard finite difference schemes. J. Phys. Conf. Ser. 490, 012205 (2014)
O.T. Kosmas, D. Papadopoulos, D. Vlachos, Geometric derivation and analysis of multi-symplectic numerical schemes for differential equations, in Computational Mathematics and Variational Analysis. Springer Optimization and Its Applications, vol. 159, ed. by N. Daras, T. Rassias (2020), pp. 207–226
O.T. Kosmas, Exponential variational integrators for the dynamics of multibody systems with holonomic constraints. J. Phys. Conf. Ser. 1391, 012170 (2019)
O.T. Kosmas, S. Leyendecker, Stability analysis of high order phase fitted variational integrators, in Proceedings of WCCM XI: ECCM V—ECFD VI, vol. 1389 (2014), pp. 865–866
A. Stern, E. Grinspun, Implicit-explicit integration of highly oscillatory problems. SIAM Multiscale Model. Simul. 7, 1779–1794 (2009)
S. Reich, Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36, 1549–1570 (1999)
M. Leok, J. Zhang, Discrete Hamiltonian variational integrators. IMA J. Numer. Anal. 31, 1497–1532 (2011)
O.T. Kosmas, D.S. Vlachos, A space-time geodesic approach for phase fitted variational integrators. J. Phys. Conf. Ser. 738, 012133 (2016)
O.T. Kosmas, Charged particle in an electromagnetic field using variational integrators. ICNAAM Numer. Anal. Appl. Math. 1389, 1927 (2011)
O.T. Kosmas, D.S. Vlachos, Local path fitting: a new approach to variational integrators. J. Comput. Appl. Math. 236, 2632–2642 (2012)
Acknowledgement
Dr. Odysseas Kosmas wishes to acknowledge the support of EPSRC via grand EP/N026136/1 “Geometric Mechanics of Solids.”
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kosmas, O., Vlachos, D. (2021). Error Analysis Through Energy Minimization and Stability Properties of Exponential Integrators. In: Rassias, T.M., Pardalos, P.M. (eds) Nonlinear Analysis and Global Optimization. Springer Optimization and Its Applications, vol 167. Springer, Cham. https://doi.org/10.1007/978-3-030-61732-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-61732-5_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-61731-8
Online ISBN: 978-3-030-61732-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)