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On embedding of subcartesian differential space and application
Backward error analysis for variational discretisations of PDEs
1. | Massey University, Private Bag 11 222, Palmerston North, 4442, New Zealand |
2. | Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany |
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby 'modified' equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.
References:
[1] |
M. Barbero-Liñán, M. F. Puiggalí, S. Ferraro and D. M. de Diego,
The inverse problem of the calculus of variations for discrete systems, Journal of Physics A: Mathematical and Theoretical, 51 (2018), 185-202.
doi: 10.1088/1751-8121/aab546. |
[2] |
P. Chartier, E. Faou and A. Murua,
An algebraic approach to invariant preserving integators: The case of quadratic and Hamiltonian invariants, Numerische Mathematik, 103 (2006), 575-590.
doi: 10.1007/s00211-006-0003-8. |
[3] |
M. E. Fels and C. G. Torre,
The principle of symmetric criticality in general relativity, Classical and Quantum Gravity, 19 (2002), 641-675.
doi: 10.1088/0264-9381/19/4/303. |
[4] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer Berlin Heidelberg, 2013.
doi: 10.1007/3-540-30666-8. |
[5] |
A. L. Islas and C. M. Schober,
Backward error analysis for multisymplectic discretizations of {H}amiltonian PDEs, Mathematics and Computers in Simulation, 69 (2005), 290-303.
doi: 10.1016/j.matcom.2005.01.006. |
[6] |
P. Libermann and C.-M. Marle, Symplectic manifolds and Poisson manifolds, in Symplectic Geometry and Analytical Mechanics, Springer Netherlands, Dordrecht, 1987, 89–184.
doi: 10.1007/978-94-009-3807-6_3. |
[7] |
E. L. Mansfield, Variational Problems with Symmetry, Cambridge Monographs on Applied
and Computational Mathematics, Cambridge University Press, 2010,206–240.
doi: 10.1017/CBO9780511844621.009. |
[8] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[9] |
F. McDonald, Travelling Wave Solutions in Multisymplectic Discretisations of Wave Equations, Ph.D thesis, Massey University, 2013. |
[10] |
F. McDonald, R. I. McLachlan, B. E. Moore and G. R. W. Quispel,
Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations, Journal of Difference Equations and Applications, 22 (2016), 913-940.
doi: 10.1080/10236198.2016.1162161. |
[11] |
R. I. McLachlan and C. Offen, Backward error analysis for conjugate symplectic methods, preprint, arXiv: 2201.03911, (2022). |
[12] |
B. Moore and S. Reich,
Backward error analysis for multi-symplectic integration methods, Numerische Mathematik, 95 (2003), 625-652.
doi: 10.1007/s00211-003-0458-9. |
[13] |
S. Ober-Blöbaum and C. Offen, Variational integration of learned dynamical systems, preprint, arXiv: 2112.12619, 2021. |
[14] |
C. Offen, GitHubrepository Christian-Offen/multisymplectic, 5 (2022)., Available from: https://github.com/Christian-Offen/multisymplectic. |
[15] |
C. Offen and S. Ober-Blöbaum,
Symplectic integration of learned Hamiltonian systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32 (2022), 013122.
doi: 10.1063/5.0065913. |
[16] |
P. J. Olver, Applications of Lie Groups to Differential Equations., Springer US, 1986.
doi: 10.1007/978-1-4684-0274-2. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
J. M. Pons,
Ostrogradski's theorem for higher-order singular lagrangians, Lett Math Phys, 17 (1989), 181-189.
doi: 10.1007/BF00401583. |
[19] |
M. S. Rashid and S. S. Khalil,
Hamiltonian description of higher order Lagrangians, International Journal of Modern Physics A, 11 (1996), 4551-4559.
doi: 10.1142/S0217751X96002108. |
[20] |
M. Vermeeren,
Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.
doi: 10.1007/s00211-017-0896-4. |
show all references
References:
[1] |
M. Barbero-Liñán, M. F. Puiggalí, S. Ferraro and D. M. de Diego,
The inverse problem of the calculus of variations for discrete systems, Journal of Physics A: Mathematical and Theoretical, 51 (2018), 185-202.
doi: 10.1088/1751-8121/aab546. |
[2] |
P. Chartier, E. Faou and A. Murua,
An algebraic approach to invariant preserving integators: The case of quadratic and Hamiltonian invariants, Numerische Mathematik, 103 (2006), 575-590.
doi: 10.1007/s00211-006-0003-8. |
[3] |
M. E. Fels and C. G. Torre,
The principle of symmetric criticality in general relativity, Classical and Quantum Gravity, 19 (2002), 641-675.
doi: 10.1088/0264-9381/19/4/303. |
[4] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, Springer Berlin Heidelberg, 2013.
doi: 10.1007/3-540-30666-8. |
[5] |
A. L. Islas and C. M. Schober,
Backward error analysis for multisymplectic discretizations of {H}amiltonian PDEs, Mathematics and Computers in Simulation, 69 (2005), 290-303.
doi: 10.1016/j.matcom.2005.01.006. |
[6] |
P. Libermann and C.-M. Marle, Symplectic manifolds and Poisson manifolds, in Symplectic Geometry and Analytical Mechanics, Springer Netherlands, Dordrecht, 1987, 89–184.
doi: 10.1007/978-94-009-3807-6_3. |
[7] |
E. L. Mansfield, Variational Problems with Symmetry, Cambridge Monographs on Applied
and Computational Mathematics, Cambridge University Press, 2010,206–240.
doi: 10.1017/CBO9780511844621.009. |
[8] |
J. E. Marsden and M. West,
Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[9] |
F. McDonald, Travelling Wave Solutions in Multisymplectic Discretisations of Wave Equations, Ph.D thesis, Massey University, 2013. |
[10] |
F. McDonald, R. I. McLachlan, B. E. Moore and G. R. W. Quispel,
Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations, Journal of Difference Equations and Applications, 22 (2016), 913-940.
doi: 10.1080/10236198.2016.1162161. |
[11] |
R. I. McLachlan and C. Offen, Backward error analysis for conjugate symplectic methods, preprint, arXiv: 2201.03911, (2022). |
[12] |
B. Moore and S. Reich,
Backward error analysis for multi-symplectic integration methods, Numerische Mathematik, 95 (2003), 625-652.
doi: 10.1007/s00211-003-0458-9. |
[13] |
S. Ober-Blöbaum and C. Offen, Variational integration of learned dynamical systems, preprint, arXiv: 2112.12619, 2021. |
[14] |
C. Offen, GitHubrepository Christian-Offen/multisymplectic, 5 (2022)., Available from: https://github.com/Christian-Offen/multisymplectic. |
[15] |
C. Offen and S. Ober-Blöbaum,
Symplectic integration of learned Hamiltonian systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32 (2022), 013122.
doi: 10.1063/5.0065913. |
[16] |
P. J. Olver, Applications of Lie Groups to Differential Equations., Springer US, 1986.
doi: 10.1007/978-1-4684-0274-2. |
[17] |
R. S. Palais,
The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
J. M. Pons,
Ostrogradski's theorem for higher-order singular lagrangians, Lett Math Phys, 17 (1989), 181-189.
doi: 10.1007/BF00401583. |
[19] |
M. S. Rashid and S. S. Khalil,
Hamiltonian description of higher order Lagrangians, International Journal of Modern Physics A, 11 (1996), 4551-4559.
doi: 10.1142/S0217751X96002108. |
[20] |
M. Vermeeren,
Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.
doi: 10.1007/s00211-017-0896-4. |








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