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Symplectic Pstable 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 IIIAB symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIAB pair, and, in addition, they are Pstable, 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), 161174. doi: 10.1016/j.cam.2016.09.047. Google Scholar 
[2] 
G. J. Cooper and A. Sayfy, Additive Runge–Kutta methods for stiff ordinary differential equations, Math. Comp., 40 (1983), 207218. doi: 10.1090/S00255718198306794411. Google Scholar 
[3] 
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, StructurePreserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2010. Google Scholar 
[4] 
L. Jay, Specialized partitioned additive Runge–Kutta methods for systems of overdetermined DAEs with holonomic constraints, SIAM J. Numer. Anal., 45 (2007), 18141842. doi: 10.1137/060667475. Google Scholar 
[5] 
L. Jay, Structure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods, SIAM J. Sci. Comput., 20 (1998), 416446. doi: 10.1137/S1064827595293223. Google Scholar 
[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. Google Scholar 
[7] 
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357514. doi: 10.1017/S096249290100006X. Google Scholar 
[8] 
R. I. McLachlan and A. Stern, Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 13781397. doi: 10.1137/130921118. Google Scholar 
[9] 
R. I. McLachlan, Y. Sun and P. S. P. Tse, Linear stability of partitioned RungeKutta methods, SIAM J. Numer. Anal., 49 (2011), 232263. doi: 10.1137/100787234. Google Scholar 
[10] 
F. Pfeil, A Higher Order IMEX Method for Solving Highly Oscillatory Problems, Master's thesis, University of Bergen, Norway, June 2019. Google Scholar 
[11] 
A. Sandu and M. Günther, A generalizedstructure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 1742. doi: 10.1137/130943224. Google Scholar 
[12] 
A. Stern and E. Grinspun, Implicitexplicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 17791794. doi: 10.1137/080732936. Google Scholar 
[13] 
G. M. Tanner, Generalized Additive Runge–Kutta Methods for Stiff Odes, PhD thesis, University of Iowa, 2018. Google Scholar 
[14] 
T. Wenger, S. OberBlöbaum 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. Google Scholar 
[15] 
T. Wenger, S. OberBlöbaum and S. Leyendecker, Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints, Adv. Comput. Math., 43 (2017), 11631195. doi: 10.1007/s1044401795205. Google Scholar 
[16] 
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262268. doi: 10.1016/03759601(90)900923. Google Scholar 
[17] 
A. Zanna, A family of modified trigonometric integrators for highly oscillatory problems, FoCM, Barcelona, 2017. Google Scholar 
[18] 
M. Zhang and R. D. Skeel, Cheap implicit symplectic integrators, Appl. Numer. Math., 25 (1997), 297302. doi: 10.1016/S01689274(97)000664. Google Scholar 
show all references
References:
[1] 
E. Celledoni and E. H. Høyseth, The averaged Lagrangian method, J. Comput. Appl. Math., 316 (2017), 161174. doi: 10.1016/j.cam.2016.09.047. Google Scholar 
[2] 
G. J. Cooper and A. Sayfy, Additive Runge–Kutta methods for stiff ordinary differential equations, Math. Comp., 40 (1983), 207218. doi: 10.1090/S00255718198306794411. Google Scholar 
[3] 
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, StructurePreserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer, 2010. Google Scholar 
[4] 
L. Jay, Specialized partitioned additive Runge–Kutta methods for systems of overdetermined DAEs with holonomic constraints, SIAM J. Numer. Anal., 45 (2007), 18141842. doi: 10.1137/060667475. Google Scholar 
[5] 
L. Jay, Structure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods, SIAM J. Sci. Comput., 20 (1998), 416446. doi: 10.1137/S1064827595293223. Google Scholar 
[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. Google Scholar 
[7] 
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357514. doi: 10.1017/S096249290100006X. Google Scholar 
[8] 
R. I. McLachlan and A. Stern, Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 13781397. doi: 10.1137/130921118. Google Scholar 
[9] 
R. I. McLachlan, Y. Sun and P. S. P. Tse, Linear stability of partitioned RungeKutta methods, SIAM J. Numer. Anal., 49 (2011), 232263. doi: 10.1137/100787234. Google Scholar 
[10] 
F. Pfeil, A Higher Order IMEX Method for Solving Highly Oscillatory Problems, Master's thesis, University of Bergen, Norway, June 2019. Google Scholar 
[11] 
A. Sandu and M. Günther, A generalizedstructure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 1742. doi: 10.1137/130943224. Google Scholar 
[12] 
A. Stern and E. Grinspun, Implicitexplicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 17791794. doi: 10.1137/080732936. Google Scholar 
[13] 
G. M. Tanner, Generalized Additive Runge–Kutta Methods for Stiff Odes, PhD thesis, University of Iowa, 2018. Google Scholar 
[14] 
T. Wenger, S. OberBlöbaum 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. Google Scholar 
[15] 
T. Wenger, S. OberBlöbaum and S. Leyendecker, Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints, Adv. Comput. Math., 43 (2017), 11631195. doi: 10.1007/s1044401795205. Google Scholar 
[16] 
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262268. doi: 10.1016/03759601(90)900923. Google Scholar 
[17] 
A. Zanna, A family of modified trigonometric integrators for highly oscillatory problems, FoCM, Barcelona, 2017. Google Scholar 
[18] 
M. Zhang and R. D. Skeel, Cheap implicit symplectic integrators, Appl. Numer. Math., 25 (1997), 297302. doi: 10.1016/S01689274(97)000664. Google Scholar 
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