Geometric numerical methods

Neglecting collisions and other dissipative effects, many models of plasma physics including kinetic, fluid, MHD and hybrid models have been shown to possess a noncanonical hamiltonian structure. This comes with some important properties in particular conservation of energy and some Casimir invariants, typically div B = 0 and also Gauss’ law for the Vlasov-Maxwell equations. Adequate conservation of these quantities has been proven to be essential for well behaving numerical solutions. Many numerical methods have been devised to this aim. Geometric numerical methods achieve this by discretizing the Hamiltonian structure, i.e. Poisson bracket and Hamiltonian, rather than the resulting PDEs. This approach approximates the infinite dimensional original hamiltonian system by a Finite Dimensional hamiltonian system and in this way guarantees the conservation of the appropriate discretized invariants. After introducing the concept of Geometric discretization, we will illustrate it on a hamiltonian formulation of the Vlasov-Maxwell model and hybrid fluid-kinetic models. We will also extend geometric numerical methods to dissipative physical systems by adding to the hamiltonian part of the model a dissipative part as for example the Landau collision operator to the Vlasov-Maxwell model in the form of a dissipative symmetric bracket yielding altogether a so-called metriplectic formulation.