Structure-preserving particle methods for collision operators in plasma physics
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Hamiltonian systems occur naturally in many physical problems, including those in plasma physics. The behaviour of charged particles in electromagnetic fields, which consistently influence each other, can be modelled through the Vlasov-Maxwell equations — a 6D time-dependent system of Hamiltonian hyperbolic partial differential equations. This Hamiltonian structure has recently been exploited in the design of numerical solvers, as it provides one way to ensure the numerical preservation of invariants associated with the problem. Introducing dissipation, such as through collisions, can spoil this pure Hamiltonian treatment. Here, the metriplectic formulation of Morrison can be used to formulate the full (Hamiltonian and dissipative) system in a geometric way. However, further complications arise when one wishes to tackle such a problem using particles, since collision operators are typically second-order differential operators. We approach this problem by $L^2$ projecting the particles onto a smoother basis. In this talk, I will present metriplectic discretisations of the Landau collision operator using particles and detail how we use projections to build our discretisations.