Structure Preserving Numerical Methods for Magnetic Fusion

Many plasma physics models have been proved to possess a non-canonical hamiltonian structure. Invariants like the hamiltonian or Casimir invariants like div B = 0 follow immediately from this structure. Hence discretizing the infinite dimensional hamiltonian structure so that we obtain a finite dimensional hamiltonian structure provides a natural way to conserve discrete invariants. After a short review of geometric discretization for kinetic and hybrid fluid-kinetic models, we will focus on structure preserving discretization of compressible ideal MHD. In order to robustly handle shocks and also keep the symmetries inherent to the MHD model, we will introduce a semi-implicit hybrid model coupling Finite Element Exterior Calculus and Finite Volume schemes. A splitting approach is designed so that we may take advantage of the conservation properties and robustness of the Finite Volume schemes for the non-linear advection, while relying on a structure-preserving discretization of the magneto-acoustic terms based on Finite Element Exterior calculus. Moreover, the nonlinear convective terms are treated via an explicit time-discretization, while the magnetic and acoustic terms are solved via an implicit time-discretization. Thanks to this, the resulting CFL condition will depend, at least formally, only on the fluid velocity and not on the Alfvén or sound speeds that may become too stringent in the low Mach regimes. In this approach, the divergence free constraint of the magnetic field is always preserved up to machine precision, and the symmetry of the physical model is also reflected to the symmetry of the final algebraic nonlinear systems that are solved in an implicit step. Thanks to the symmetry of the systems, the very efficient matrix-free conjugate gradient method may be employed.