# American Institute of Mathematical Sciences

September  2022, 14(3): 447-471. doi: 10.3934/jgm.2022014

## 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

*Corresponding author: Christian Offen

Received  January 2022 Published  September 2022 Early access  June 2022

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.

Citation: Robert I McLachlan, Christian Offen. Backward error analysis for variational discretisations of PDEs. Journal of Geometric Mechanics, 2022, 14 (3) : 447-471. doi: 10.3934/jgm.2022014
##### 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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##### 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.
Illustration of Theorem 1.1. The left hand column gives the actions of a PDE and an associated ODE that governs its symmetric solutions such as travelling waves. The right hand column gives three Lagrangians of modified equations of a variational discretization. Top: of the discretization, containing arbitrarily high derivatives; middle: of its symmetric solutions, containing arbitrarily high derivatives; and bottom: of its symmetric solutions, containing first derivatives only. $\tilde{\mathcal L}$ can be regarded as a modified Lagrangian of $L^0$. ${\mathcal L}_\Delta$, $\mathcal L$ and $\tilde {\mathcal L}$ are formal power series in the step sizes
Dynamics of the amplitude variable $\phi_1(\xi)$ for $\alpha \in \{0, 0.3, 0.5, 0.7\}$ for $V(a) = -\exp(-(a-1)^2)$ and the wave speed $c = 0.5$. (Initial condition $\phi_1(0) = \phi_2(0) = \dot\phi_1(0) = \dot\phi_2(0) = 0.1$)
Phase portrait of the amplitude variables $\phi_1(\xi)$, $\phi_2(\xi)$ for $\alpha \in \{0, 0.1, 0.6\}$, $V(a) = -\exp(-(a-1)^2)$, the wave speed $c = 0.5$ and $\xi \in [-5, 10]$. (Initial condition $\phi_1(0) = \phi_2(0) = \dot\phi_1(0) = \dot\phi_2(0) = 0.1$)
Evaluation of the conserved quantity $I_{\mathrm{rot}}$ (see 12) along a numerically computed trajectory shows round-off errors only (vertical axis is scaled by $10^{-14}$). Here $V(a) = -\frac 12 a -a^2$, $\alpha = -1$, $c = 2$. The integrator is the symplectic midpoint rule. Implicit equations are solved using fixed point iterations
Dynamics of the amplitude variable $\phi_1(\xi)$ for $\alpha = 0$, $V(a) = -\exp(-(a-1)^2)$, $c = 0.5$ and $\Delta x \in \{0, 0.6, 1, 1.2\}$ for the modified equation truncated after $\mathcal O(h^3)$ terms.
Numerical integration of the ODE (27) truncated after $\mathcal O(h^2)$ terms with $V(s) = -0.1 s^4 +s$, $\Delta x = 0.1$, $\Delta t = 0.15$, $\alpha = 0.3$, $c = 2$. All numerical computations have been performed in the Darboux variables $(\mathfrak q, \mathfrak p)$ of the continuous system using the implicit midpoint rule combined with fixed-point iterations. Therefore, the integration is symplectic modulo second order terms. The plots show a phase plot of a motion initialised at $(\mathfrak q, \mathfrak p) = (-0.11, -0.01, -0.1, 0.1)$ and the behaviour of the Hamiltonian $H$ of the exact system and the Hamiltonian $\mathcal H$ of the modified system truncated after $\mathcal O(h^2)$ terms as well as the behaviour of the conserved quantity of the exact system $I_{\mathrm{rot}}$ and of the modified system $I_{\mathrm{rot}}^{\mathrm{mod}}$ truncated after $\mathcal O(h^2)$ terms along the motion
When $c \Delta t/\Delta x$ is rational, the functional equation (25) yields a multistep formula. The series parameter $h$ is set to 1. We use $V(s) = s^2$, $\Delta t = 0.15$, $\Delta x = 2 c \Delta t$. Let $\Delta \tau = c \Delta t$. To initialise the scheme, values at $\xi = \Delta \tau, 2\Delta \tau, 3\Delta \tau$ are obtained by integrating (27) truncated to 4th order with high accuracy with the initial condition $(\phi(0), \dot{\phi}(0)) = ((0.1, -0.05), (0, 0.1))$
Interpretation of (4) as a multistep formula for $\frac mn = \frac{\Delta x}{c\Delta t}<1$. The variable $\hat \xi$ corresponds to $\xi - c \Delta t$ when comparing with (4) and $\Delta s = 2 c \Delta t$
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