Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws

Abstract

International audienceA reduction method is presented for systems of conservation laws with boundary energy flow. It is stated as a generalized pseudo-spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port-variables. The symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws, for any choice of collocation points (i.e. for any polynomial bases). The symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum, on the example of the ideal transmission line, is discussed in comparison with the suggested reduction scheme

A port-Hamiltonian approach to modeling the structural dynamics of complex systems

“…For an accurate approximation, a relatively large number of finite elements is usually required with these approaches. Moulla et al [13] adopted a collocation method to discretize 1D port-Hamiltonian systems. Using polynomial bases, this results in higher accuracy for lower-order approximations.…”

A Partitioned Finite Element Method for the Structure-Preserving Discretization of Damped Infinite-Dimensional Port-Hamiltonian Systems with Boundary Control

“…p , e r ) , e ∂ := u ∂ , f ∂ := −y ∂ }, is a Dirac structure. Remark 1. : Moreover, contrarily to other structure-preserving methods relying on Stokes-Dirac structure, like [20,13], there is no need here to project, reduce, some non square matrices in order to recover a full rank system at the discrete level, which is, at least from the numerical point of view, a severe limitation indeed.…”

“…Our main concern is to provide a numerical method that preserves, at the discrete level, the geometrical structure of the original controlled PDE; for short, we look for a structure-preserving method which automatically transforms the Stokes-Dirac structure into a finite-dimensional Dirac structure: in the last decade, quite a number of ways have been explored, see e.g. [20,26,13,10,19]. Recently in [4], a method based on the weak formulation of the Partial Differential Equation and the use of the celebrated Finite Element Method has emerged.…”

Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates

“…In [13], the authors made use of a mixed finite element spatial discretization for 1D hyperbolic system of conservation laws, introducing different low-order basis functions for the energy and co-energy variables. Pseudo-spectral methods relying on higher-order global polynomial approximations were studied in [14]. This method was used and extended to take into account unbounded control operators in [15].…”