Geometric discretization for a plasma control model

we define a family of geometric discretization methods for the reduction of a 1D distributed parameters systems of conservation laws and apply these methods to the reduction of plasma control model written in Port-Controlled Hamiltonian (PCH) form. In these discrete schemes, variables are projected into appropriate bases in order to perform exact spatial differentiation. We show that some spectral and energetical properties are therefore preserved. A geometric (symplectic) collocation scheme using Lagrange polynomials is investigated. Numerical results show oscillations in the transient response in case of non homogeneous boundary conditions or sharp distributed control. A second symplectic spectral scheme using Bessel conjugated bases is then derived which allows a more accurate approximation of eigenfunctions and reduces the unwanted numerical oscillations. Finally, the proposed numerical integration of the control model is validated against experimental data from the tokamak Tore Supra.