American Institute of Mathematical Sciences

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

Symplectic integration of PDEs using Clebsch variables

Robert I McLachlan 1, , Christian Offen 1, and  Benjamin K Tapley 2,

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.

Keywords: Symplectic integration, Lie-Poisson system, Burgers' equation, Euler's equation.
Mathematics Subject Classification: Primary: 37M15, 65P10; Secondary: 37K05, 35Q31, 37J15, 53D20.
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:
[1]

L. Brugnano, M. Calvo, J. Montijano and L. Rández, Energy-preserving methods for Poisson systems, Journal of Computational and Applied Mathematics, 236 (2012), 3890–3904, 40 years of numerical analysis: "Is the discrete world an approximation of the continuous one or is it the other way around". doi: 10.1016/j.cam.2012.02.033.

[2]

A. Chern, F. Knöppel, U. Pinkall, P. Schröder and S. Weiẞmann, Schrödinger's smoke, ACM Transactions on Graphics (TOG), 35 (2016), 77. doi: 10.1145/2897824.2925868.

[3]

D. Cohen and E. Hairer, Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51 (2011), 91-101.  doi: 10.1007/s10543-011-0310-z.

[4]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 305205, 14pp. doi: 10.1088/1751-8113/44/30/305205.

[5]

D. M. de Diego, Lie-Poisson integrators, preprint, arXiv: 1803.01427, URL https://arXiv.org/abs/1803.01427.

[6]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[7]

B. Khesin and R. Wendt, Infinite-dimensional lie groups: Their geometry, orbits, and dynamical systems, The Geometry of Infinite-Dimensional Groups, 2009, 47–153. doi: 10.1007/978-3-540-77263-7_2.

[8]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, vol. 53, AMS, 1997. doi: 10.1090/surv/053.

[9]

E. Kuznetsov and A. Mikhailov, On the topological meaning of canonical Clebsch variables, Physics Letters A, 77 (1980), 37-38.  doi: 10.1016/0375-9601(80)90627-1.

[10]

J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1980), 305-323.  doi: 10.1016/0167-2789(83)90134-3.

[11]

J. E. Marsden and R. Abraham, Foundations of Mechanics, 2nd edition, Addison-Wesley Publishing Co., Redwood City, CA., 1978, URL http://resolver.caltech.edu/CaltechBOOK:1987.001.

[12]

J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer New York, New York, NY, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

R. I. McLachlan, Spatial discretization of partial differential equations with integrals, IMA Journal of Numerical Analysis, 23 (2003), 645-664.  doi: 10.1093/imanum/23.4.645.

[15]

R. I. McLachlan, K. Modin and O. Verdier, Collective symplectic integrators, Nonlinearity, 27 (2014), 1525-1542.  doi: 10.1088/0951-7715/27/6/1525.

[16]

I. Vaisman, Symplectic realizations of poisson manifolds, Lectures on the Geometry of Poisson Manifolds, 1994,115–133. doi: 10.1007/978-3-0348-8495-2_9.

[17]

C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 030, 22 pp. doi: 10.3842/SIGMA.2008.030.

show all references

References:
[1]

L. Brugnano, M. Calvo, J. Montijano and L. Rández, Energy-preserving methods for Poisson systems, Journal of Computational and Applied Mathematics, 236 (2012), 3890–3904, 40 years of numerical analysis: "Is the discrete world an approximation of the continuous one or is it the other way around". doi: 10.1016/j.cam.2012.02.033.

[2]

A. Chern, F. Knöppel, U. Pinkall, P. Schröder and S. Weiẞmann, Schrödinger's smoke, ACM Transactions on Graphics (TOG), 35 (2016), 77. doi: 10.1145/2897824.2925868.

[3]

D. Cohen and E. Hairer, Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51 (2011), 91-101.  doi: 10.1007/s10543-011-0310-z.

[4]

M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 305205, 14pp. doi: 10.1088/1751-8113/44/30/305205.

[5]

D. M. de Diego, Lie-Poisson integrators, preprint, arXiv: 1803.01427, URL https://arXiv.org/abs/1803.01427.

[6]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Second edition. Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[7]

B. Khesin and R. Wendt, Infinite-dimensional lie groups: Their geometry, orbits, and dynamical systems, The Geometry of Infinite-Dimensional Groups, 2009, 47–153. doi: 10.1007/978-3-540-77263-7_2.

[8]

A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, vol. 53, AMS, 1997. doi: 10.1090/surv/053.

[9]

E. Kuznetsov and A. Mikhailov, On the topological meaning of canonical Clebsch variables, Physics Letters A, 77 (1980), 37-38.  doi: 10.1016/0375-9601(80)90627-1.

[10]

J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1980), 305-323.  doi: 10.1016/0167-2789(83)90134-3.

[11]

J. E. Marsden and R. Abraham, Foundations of Mechanics, 2nd edition, Addison-Wesley Publishing Co., Redwood City, CA., 1978, URL http://resolver.caltech.edu/CaltechBOOK:1987.001.

[12]

J. E. Marsden, S. Pekarsky and S. Shkoller, Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.  doi: 10.1088/0951-7715/12/6/314.

[13]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer New York, New York, NY, 1999. doi: 10.1007/978-0-387-21792-5.

[14]

R. I. McLachlan, Spatial discretization of partial differential equations with integrals, IMA Journal of Numerical Analysis, 23 (2003), 645-664.  doi: 10.1093/imanum/23.4.645.

[15]

R. I. McLachlan, K. Modin and O. Verdier, Collective symplectic integrators, Nonlinearity, 27 (2014), 1525-1542.  doi: 10.1088/0951-7715/27/6/1525.

[16]

I. Vaisman, Symplectic realizations of poisson manifolds, Lectures on the Geometry of Poisson Manifolds, 1994,115–133. doi: 10.1007/978-3-0348-8495-2_9.

[17]

C. Vizman, Geodesic equations on diffeomorphism groups, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 030, 22 pp. doi: 10.3842/SIGMA.2008.030.

Figure 1.  Uniform periodic grids on $ S^1 \cong \mathbb{R}/L\mathbb{Z} $, $ L>0 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The errors corresponding to the conventional () and collective () methods for the inviscid Burgers' equation and $ \mathcal{O}(t^2) $ reference lines ()
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 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) $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  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.
Table Options

Download as excel

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]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[2]

Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55
[3]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292
[4]

Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769
[5]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121
[6]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793
[7]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341
[8]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[9]

Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835
[10]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure and Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391
[11]

Alexandre Boritchev. Decaying turbulence for the fractional subcritical Burgers equation. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2229-2249. doi: 10.3934/dcds.2018092
[12]

Naoki Fujino, Mitsuru Yamazaki. Burgers' type equation with vanishing higher order. Communications on Pure and Applied Analysis, 2007, 6 (2) : 505-520. doi: 10.3934/cpaa.2007.6.505
[13]

Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683
[14]

Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951
[15]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[16]

Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847
[17]

Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure and Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012
[18]

Wasim Akram, Debanjana Mitra. Local stabilization of viscous Burgers equation with memory. Evolution Equations and Control Theory, 2022, 11 (3) : 939-973. doi: 10.3934/eect.2021032
[19]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations and Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
[20]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511

 Impact Factor: 

Tools

Metrics

  • PDF downloads (185)
  • HTML views (634)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy