Asymptotically Stable Particle-in-Cell Methods for the Magnetized Vlasov–Poisson Equations in Orthogonal Curvilinear Coordinates

In high-temperature plasma physics, strong magnetic fields are essential for confining charged particles. Consequently, classical mathematical models of such systems must take into account the external magnetic field effects. A key governing equation is the magnetized Vlasov–Poisson system, which exhibits multiscale dynamics and rich physical properties. Thus, developing the structure-preserving numerical methods and studying its capability of maintaining these intrinsic properties over long-time simulations is therefore critically important. This paper presents a general framework for constructing and analyzing structure-preserving methods in orthogonal curvilinear coordinates. We prove that in these coordinates the Poisson-bracket structure is retained under appropriate finite element discretizations. However, the resulting Hamiltonian systems in transformed coordinates typically can’t be decomposed as several subsystems which can be solved exactly. To address this, we propose a semi-implicit numerical scheme that can still maintain the favorable stability properties inherited by the system. The effectiveness of the new derived numerical methods is demonstrated in application to strongly magnetized systems, and the rigorous asymptotic stability analysis are provided.