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In high-temperature plasma physics, a strong magnetic field is usually usedto confine charged particles. Therefore, for studying the classicalmathematical models of the physical problems it needs to consider the effect ofexternal magnetic fields. One of the important model equations in plasma is theVlasov-Poisson equation with an external magnetic field. This equation usuallyhas multi-scale characteristics and rich physical properties, thus it is veryimportant and meaningful to construct numerical methods that can maintain thephysical properties inherited by the original systems over long time. Thispaper extends the corresponding theory in Cartesian coordinates to generalorthogonal curvilinear coordinates, and proves that a Poisson-bracket structurecan still be obtained after applying the corresponding finite elementdiscretization. However, the Hamiltonian systems in the new coordinate systemsgenerally cannot be decomposed into sub-systems that can be solved accurately,so it is impossible to use the splitting methods to construct the correspondinggeometric integrators. Therefore, this paper proposes a semi-implicit methodfor strong magnetic fields and analyzes the asymptotic stability of thismethod.

Asymptotically stable Particle-in-Cell methods for the magnetized Vlasov--Poisson equations in orthogonal curvilinear coordinates