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An algebraic approach to the spontaneous formation of spherical jets
Symplectic P-stable additive Runge—Kutta methods
Department of Mathematics, University of Bergen, Postbox 7803, 5020 Bergen, Norway |
Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are P-stable, therefore suitable for application to highly oscillatory problems.
References:
[1] |
E. Celledoni and E. H. Høyseth,
The averaged Lagrangian method, J. Comput. Appl. Math., 316 (2017), 161-174.
doi: 10.1016/j.cam.2016.09.047. |
[2] |
G. J. Cooper and A. Sayfy,
Additive Runge–Kutta methods for stiff ordinary differential equations, Math. Comp., 40 (1983), 207-218.
doi: 10.1090/S0025-5718-1983-0679441-1. |
[3] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2010. |
[4] |
L. Jay,
Specialized partitioned additive Runge–Kutta methods for systems of overdetermined DAEs with holonomic constraints, SIAM J. Numer. Anal., 45 (2007), 1814-1842.
doi: 10.1137/060667475. |
[5] |
L. Jay,
Structure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods, SIAM J. Sci. Comput., 20 (1998), 416-446.
doi: 10.1137/S1064827595293223. |
[6] |
L. O. Jay and L. R. Petzold, Highly oscillatory systems and periodic stability, Technical Report, Army High Performance Computing Research Center, Stanford, CA, (1995), 95–105. |
[7] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[8] |
R. I. McLachlan and A. Stern,
Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 1378-1397.
doi: 10.1137/130921118. |
[9] |
R. I. McLachlan, Y. Sun and P. S. P. Tse,
Linear stability of partitioned Runge-Kutta methods, SIAM J. Numer. Anal., 49 (2011), 232-263.
doi: 10.1137/100787234. |
[10] |
F. Pfeil, A Higher Order IMEX Method for Solving Highly Oscillatory Problems, Master's thesis, University of Bergen, Norway, June 2019. |
[11] |
A. Sandu and M. Günther,
A generalized-structure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 17-42.
doi: 10.1137/130943224. |
[12] |
A. Stern and E. Grinspun,
Implicit-explicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 1779-1794.
doi: 10.1137/080732936. |
[13] |
G. M. Tanner, Generalized Additive Runge–Kutta Methods for Stiff Odes, PhD thesis, University of Iowa, 2018. |
[14] |
T. Wenger, S. Ober-Blöbaum and S. Leyendecker, Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials, ECCOMAS Congress 2016, 2016.
doi: 10.7712/100016.1920.10163. |
[15] |
T. Wenger, S. Ober-Blöbaum and S. Leyendecker,
Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints, Adv. Comput. Math., 43 (2017), 1163-1195.
doi: 10.1007/s10444-017-9520-5. |
[16] |
H. Yoshida,
Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.
doi: 10.1016/0375-9601(90)90092-3. |
[17] |
A. Zanna, A family of modified trigonometric integrators for highly oscillatory problems, FoCM, Barcelona, 2017. |
[18] |
M. Zhang and R. D. Skeel,
Cheap implicit symplectic integrators, Appl. Numer. Math., 25 (1997), 297-302.
doi: 10.1016/S0168-9274(97)00066-4. |
show all references
References:
[1] |
E. Celledoni and E. H. Høyseth,
The averaged Lagrangian method, J. Comput. Appl. Math., 316 (2017), 161-174.
doi: 10.1016/j.cam.2016.09.047. |
[2] |
G. J. Cooper and A. Sayfy,
Additive Runge–Kutta methods for stiff ordinary differential equations, Math. Comp., 40 (1983), 207-218.
doi: 10.1090/S0025-5718-1983-0679441-1. |
[3] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2010. |
[4] |
L. Jay,
Specialized partitioned additive Runge–Kutta methods for systems of overdetermined DAEs with holonomic constraints, SIAM J. Numer. Anal., 45 (2007), 1814-1842.
doi: 10.1137/060667475. |
[5] |
L. Jay,
Structure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods, SIAM J. Sci. Comput., 20 (1998), 416-446.
doi: 10.1137/S1064827595293223. |
[6] |
L. O. Jay and L. R. Petzold, Highly oscillatory systems and periodic stability, Technical Report, Army High Performance Computing Research Center, Stanford, CA, (1995), 95–105. |
[7] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[8] |
R. I. McLachlan and A. Stern,
Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 1378-1397.
doi: 10.1137/130921118. |
[9] |
R. I. McLachlan, Y. Sun and P. S. P. Tse,
Linear stability of partitioned Runge-Kutta methods, SIAM J. Numer. Anal., 49 (2011), 232-263.
doi: 10.1137/100787234. |
[10] |
F. Pfeil, A Higher Order IMEX Method for Solving Highly Oscillatory Problems, Master's thesis, University of Bergen, Norway, June 2019. |
[11] |
A. Sandu and M. Günther,
A generalized-structure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 17-42.
doi: 10.1137/130943224. |
[12] |
A. Stern and E. Grinspun,
Implicit-explicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 1779-1794.
doi: 10.1137/080732936. |
[13] |
G. M. Tanner, Generalized Additive Runge–Kutta Methods for Stiff Odes, PhD thesis, University of Iowa, 2018. |
[14] |
T. Wenger, S. Ober-Blöbaum and S. Leyendecker, Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials, ECCOMAS Congress 2016, 2016.
doi: 10.7712/100016.1920.10163. |
[15] |
T. Wenger, S. Ober-Blöbaum and S. Leyendecker,
Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints, Adv. Comput. Math., 43 (2017), 1163-1195.
doi: 10.1007/s10444-017-9520-5. |
[16] |
H. Yoshida,
Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.
doi: 10.1016/0375-9601(90)90092-3. |
[17] |
A. Zanna, A family of modified trigonometric integrators for highly oscillatory problems, FoCM, Barcelona, 2017. |
[18] |
M. Zhang and R. D. Skeel,
Cheap implicit symplectic integrators, Appl. Numer. Math., 25 (1997), 297-302.
doi: 10.1016/S0168-9274(97)00066-4. |

















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