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Lie-Poisson methods for isospectral flows and their application to long-time simulation of spherical ideal hydrodynamics

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GCS - Geometry, compatibility and structure preservation in computational differential equations

The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie–Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectra in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie-Poisson structure. Here we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie-Poisson structure. The methods are surprisingly simple, and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie–Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to long-time simulation of the Euler equations on a sphere. Our findings suggest that our structure-preserving algorithms, on the one hand, perform at least as well as other popular methods (i.e. CLAM ) without adding spurious hyperviscosity terms, on the other hand, show that the conservation of the Casimir functions can be actually used to predict the final state of the fluid




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