# American Institute of Mathematical Sciences

June  2019, 6(1): 111-130. doi: 10.3934/jcd.2019005

## Symplectic integration of PDEs using Clebsch variables

 1 School of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, 4442, New Zealand 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Gløshaugen, Norway

Published  July 2019

Fund Project: This research was supported by the Marsden Fund of the Royal Society Te Apārangi.

Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.

Citation: Robert I McLachlan, Christian Offen, Benjamin K Tapley. Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics, 2019, 6 (1) : 111-130. doi: 10.3934/jcd.2019005
##### References:

show all references

##### References:
Uniform periodic grids on $S^1 \cong \mathbb{R}/L\mathbb{Z}$, $L>0$
Order-two convergence for the travelling wave solution of the extended Burgers' equation outlined in section 6.2. The plots correspond to the conventional solution (○) and the collective solution (△) and an order-two reference line (). The error is calculated after 512 timesteps, with $L = 8$, $\Delta t = 2^{-14}$ and $\Delta x = L/2^{k}$ for $k = 1,2,3$ and $4$
Inviscid Burgers' equation solutions of the conventional method () and collective method (). The grid parameters are $n_x = 64$, $\Delta x = 0.125$, $L = 8$ and $\Delta t = 2^{-12}$. A shock forms at about $t = 0.4$
The errors corresponding to the conventional () and collective () methods for the inviscid Burgers' equation and $\mathcal{O}(t^2)$ reference lines ()
Travelling wave solutions of the perturbed Burgers' equation (top row) and the positive Fourier modes (bottom row) at $t = 109$ (left column), $t = 218$ (middle column) and $t = 437$ (right column). The plots correspond to the conventional method (), collective method () and the exact travelling wave solution (). The grid parameters are $n_x = 16$, $\Delta x = 0.5$, $L = 8$ and $\Delta t = 2^{-6}$
The errors corresponding to the conventional () and collective () methods for the travelling wave experiment. The reference lines () are $\mathcal{O}(t)$ in figures (a) and (b) and exponential in figure (c)
Periodic bump solutions of the extended Burgers' equation (top row) and the positive Fourier modes (bottom row) at $t = 10$ (left column), $t = 100$ (middle column) and $t = 1000$ (right column). The plots correspond to the conventional method () and the collective method (). The grid parameters are $n_x = 32$, $\Delta x = 0.25$, $L = 8$ and $\Delta t = 2^{-8}$
The errors corresponding to the conventional () and collective () methods for the periodic bump example. The reference line () in figure (a) is $\mathcal{O}(t)$
Overview of the setting
 Continuous system Spatially discretised system Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables $q_t = \frac{\delta \bar H}{\delta p}, \quad p_t = -\frac{\delta \bar H}{\delta q}.$ Exact solutions preserve the symplectic structure, the Hamiltonian $\bar H=H\circ J$, all quantities related to the Casimirs of the original PDE and the fibres of the Clebsch map $J(q,p)=u$. Canonical Hamiltonian ODEs in $2N$ variables $\hat q_t = \nabla_{\hat p} \hat {\bar H}, \quad \hat p_t = - \nabla_{\hat q} \hat {\bar H}.$ The exact flow preserves the symplectic structure and the Hamiltonian $\hat {\bar H}$. Time-integration with the midpoint rule is symplectic. Original PDE, interpreted as a Lie-Poisson equation $u_t = \mathrm{ad}^\ast_{\frac {\delta H}{\delta u}}u.$ Exact solutions preserve the Poisson structure, the Hamiltonian $H$ and all Casimirs. Non-Hamiltonian ODEs in $N$ variables $\hat u_t = K(\hat u) \nabla_{\hat u} \hat H, \qquad K^T=-K.$ Exact solutions conserve $\hat H$. Time-integration with the midpoint rule is not symplectic.
 Continuous system Spatially discretised system Collective Hamiltonian system on an infinite-dimensional symplectic vector space in Clebsch variables $q_t = \frac{\delta \bar H}{\delta p}, \quad p_t = -\frac{\delta \bar H}{\delta q}.$ Exact solutions preserve the symplectic structure, the Hamiltonian $\bar H=H\circ J$, all quantities related to the Casimirs of the original PDE and the fibres of the Clebsch map $J(q,p)=u$. Canonical Hamiltonian ODEs in $2N$ variables $\hat q_t = \nabla_{\hat p} \hat {\bar H}, \quad \hat p_t = - \nabla_{\hat q} \hat {\bar H}.$ The exact flow preserves the symplectic structure and the Hamiltonian $\hat {\bar H}$. Time-integration with the midpoint rule is symplectic. Original PDE, interpreted as a Lie-Poisson equation $u_t = \mathrm{ad}^\ast_{\frac {\delta H}{\delta u}}u.$ Exact solutions preserve the Poisson structure, the Hamiltonian $H$ and all Casimirs. Non-Hamiltonian ODEs in $N$ variables $\hat u_t = K(\hat u) \nabla_{\hat u} \hat H, \qquad K^T=-K.$ Exact solutions conserve $\hat H$. Time-integration with the midpoint rule is not symplectic.
 [1] Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299 [2] Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 [3] Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233 [4] Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003 [5] Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 [6] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 [7] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [8] Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303 [9] Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 [10] Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124 [11] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 [12] Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025 [13] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447 [14] Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011 [15] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [16] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [17] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [18] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [19] Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370 [20] François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

Impact Factor: