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Articles

Multisymplectic Hamiltonian variational integrators

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Pages 113-157 | Received 01 Feb 2021, Accepted 20 Oct 2021, Published online: 16 Nov 2021

References

  • R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, MA, 1978. Second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman.
  • D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), pp. 1–155.
  • D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc 47(2) (2010), pp. 281–354.
  • G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys. 74 (1994), pp. 1117–1143.
  • T.J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc.121(1) (1997), pp. 147–190.
  • T.J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284(4) (2001), pp. 184–193.
  • T.J. Bridges and S. Reich, Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations, Phys. D 152–153 (2001), pp. 491–504. Advances in nonlinear mathematics and science: A special issue to honor Vladimir Zakharov.
  • T.J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A 39(19) (2006), pp. 5287–5320.
  • J. Chen, Variational formulation for the multisymplectic Hamiltonian systems, Lett. Math. Phys.71(3) (2005), pp. 243–253.
  • J.-B. Chen, Variational integrators and the finite element method, Appl. Math. Comput. 196(2) (2008), pp. 941–958.
  • F. Demoures, F. Gay-Balmaz, and T.S. Ratiu, Multisymplectic variational integrators for nonsmooth Lagrangian cntinuum mechanics, Forum Math. Sigma 4 (2016), e19.
  • V. Duruisseaux, J. Schmitt, and M. Leok, Adaptive Hamiltonian variational integrators and symplectic accelerated optimization. SIAM J. Sci. Comput. 43(4) (2021), pp. A2949–A2980.
  • M.J. Gotay, J. Isenberg, J.E. Marsden, and R. Montgomery, Momentum maps and classical relativistic fields. Part I: Covariant field theory, preprint (1998). Available at arXiv:physics/9801019 [math-ph].
  • M.J. Gotay, J. Isenberg, J.E. Marsden, and R. Montgomery, Momentum maps and classical relativistic fields. Part II: Canonical analysis of field theories, preprint (2004). Available at arXiv:physics/0411032[math-ph].
  • E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math. 25(2–3) (1997), pp. 219–227.
  • E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2006.
  • J. Hall and M. Leok, Spectral variational integrators, Numer. Math. 130(4) (2015), pp. 681–740.
  • R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), pp. 237–339.
  • J. Hong, H. Liu, and G. Sun, The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs, Math. Comp. 75 (2006), pp. 167–181.
  • A.L. Islas and C.M. Schober, On the preservation of phase space structure under multisymplectic discretization, J. Comp. Phys. 197(2) (2004), pp. 585–609.
  • A.L. Islas and C.M. Schober, Conservation properties of multisymplectic integrators, Future Gener. Comput. Syst. 22(4) (2006), pp. 412–422.
  • S. Lall and M. West, Discrete variational Hamiltonian mechanics, J. Phys. A 39(19) (2006), pp. 5509–5519.
  • T. Leitz, R.T. Sato Martin de Aimagro, and S. Leyendecker, Multisymplectic Galerkin Lie group variational integrators for geometrically exact beam dynamics based on unit dual quaternion interpolation, Comput. Methods Appl. Mech. Eng. 374 (2021), Article ID 113475.
  • M. Leok, Variational discretizations of gauge field theories using group-equivariant interpolation, Found. Comput. Math. 19(5) (2019), pp. 965–989.
  • M. Leok and T. Ohsawa, Variational and geometric structures of discrete Dirac mechanics, Found. Comput. Math. 11(5) (2011), pp. 529–562.
  • M. Leok and J. Zhang, Discrete Hamiltonian variational integrators, IMA J. Numer. Anal. 31(4) (2011), pp. 1497–1532.
  • M. León, P.D. Prieto-Martínez, N. Román-Roy, and S. Vilariño, Hamilton-Jacobi theory in multisymplectic classical field theories, J. Math. Phys. 58 (2017), Article ID 092901, 36 pp.
  • A. Lew, J.E. Marsden, M. Ortiz, and M. West, Asynchronous variational integrators, Arch. Ration. Mech. Anal. 167(2) (2003), pp. 85–146.
  • J.E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians, and water waves, Math. Proc. Cambridge Philos. Soc. 125(3) (1999), pp. 553–575.
  • J.E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer. 10 (2001), pp. 357–514.
  • J.E. Marsden, G.W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys. 199(2) (1998), pp. 351–395.
  • J.E. Marsden, S. Pekarsky, S. Shkoller, and M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys. 38(3–4) (2001), pp. 253–284.
  • R.I. McLachlan and A. Stern, Multisymplecticity of hybridizable discontinuous Galerkin methods, Found. Comput. Math. 20(1) (2020), pp. 35–69.
  • S. Ober-Blöbaum, Galerkin variational integrators and modified symplectic Runge–Kutta methods, IMA J. Numer. Anal. 37(1) (February 2016), pp. 375–406.
  • T. Ohsawa, A.M. Bloch, and M. Leok, Discrete Hamilton–Jacobi theory and discrete optimal control, in Proceedings of the IEEE Conference on Decision and Control, 2010, pp. 5438–5443.
  • S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36 (1999), pp. 1549–1570.
  • S. Reich, Finite volume methods for multi-symplectic PDEs, BIT Numer. Math. 40(3) (2000), pp. 559–582.
  • S. Reich, Multi-symplectic Runge–Kutta collocation methods for Hamiltonian wave equations, J. Comp. Phys. 157(2) (2000), pp. 473–499.
  • B.N. Ryland, R.I. Mclachlan, and J. Frank, On the multisymplecticity of partitioned Runge–Kutta and splitting methods, Int. J. Comput. Math. 84(6) (2007), pp. 847–869.
  • J.M. Schmitt and M. Leok, Properties of Hamiltonian variational integrators, IMA J. Numer. Anal.36(2) (2018), pp. 377–398.
  • J.M. Schmitt, T. Shingel, and M. Leok, Lagrangian and Hamiltonian Taylor variational integrators, BIT Numer. Math. 58(2) (2018), pp. 457–488.
  • Y.-F. Tang, Formal energy of a symplectic scheme for Hamiltonian systems and its applications (I), Comput. Math. Appl. 27(7) (1994), pp. 31–39.
  • J. Vankerschaver, H. Yoshimura, and M. Leok, The Hamilton–Pontryagin principle and multi-Dirac structures for classical field theories, J. Math. Phys. 53(7) (2012), Article ID 072903 (25 pages).
  • J. Vankerschaver, C. Liao, and M. Leok, Generating functionals and Lagrangian partial differential equations, J. Math. Phys. 54(8) (2013), Article ID 082901 (22 pages).

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