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An algebraic approach to the spontaneous formation of spherical jets
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. 
[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. 
[3] 
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, StructurePreserving 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), 18141842. 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), 416446. 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), 357514. doi: 10.1017/S096249290100006X. 
[8] 
R. I. McLachlan and A. Stern, Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 13781397. doi: 10.1137/130921118. 
[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. 
[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 generalizedstructure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 1742. doi: 10.1137/130943224. 
[12] 
A. Stern and E. Grinspun, Implicitexplicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 17791794. 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. 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. 
[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. 
[16] 
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262268. doi: 10.1016/03759601(90)900923. 
[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), 297302. doi: 10.1016/S01689274(97)000664. 
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. 
[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. 
[3] 
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, StructurePreserving 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), 18141842. 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), 416446. 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), 357514. doi: 10.1017/S096249290100006X. 
[8] 
R. I. McLachlan and A. Stern, Modified trigonometric integrators, SIAM J. Numer. Anal., 52 (2014), 13781397. doi: 10.1137/130921118. 
[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. 
[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 generalizedstructure approach to additive Runge–Kutta methods, SIAM J. Numer. Anal., 53 (2015), 1742. doi: 10.1137/130943224. 
[12] 
A. Stern and E. Grinspun, Implicitexplicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), 17791794. 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. 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. 
[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. 
[16] 
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262268. doi: 10.1016/03759601(90)900923. 
[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), 297302. doi: 10.1016/S01689274(97)000664. 
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