Article Contents
Article Contents

Symplectic integration of PDEs using Clebsch variables

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.

Mathematics Subject Classification: Primary: 37M15, 65P10; Secondary: 37K05, 35Q31, 37J15, 53D20.

 Citation:

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

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

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 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 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 8.  The errors corresponding to the conventional () and collective () methods for the periodic bump example. The reference line () in figure (a) is $\mathcal{O}(t)$

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.
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