Symplectic integration of PDEs using Clebsch variables
For researchers in geometric numerical integration, this provides a method to apply symplectic integrators to PDEs with Lie-Poisson structure, though the approach is demonstrated only on specific examples and may be incremental.
The authors lift PDEs with Lie-Poisson structure (e.g., Burgers' equation) to a symplectic manifold using Clebsch variables, enabling symplectic integration. Numerical results show excellent conservation properties, suggesting the increased phase-space dimension is offset by the benefits of symplectic integration.
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.