A modified Crank-Nicolson scheme for the Vlasov-Poisson system with a strong external magnetic field

We propose and study a Particle-In-Cell (PIC) method utilizing Crank-Nicolson time discretization for the Vlasov-Poisson system with a strong, inhomogeneous external magnetic field with fixed direction. Our focus is on particle dynamics in the plane orthogonal to the magnetic field. In this regime, traditional explicit schemes are constrained by stability conditions linked to the small Larmor radius and plasma frequency [1]. To avoid this limitation, our approach is based on numerical schemes [2, 3, 4], providing a consistent PIC discretization of the guiding-center system taking into account variations of the magnetic field. We carry out some theoretical proofs and perform several numerical experiments to validate the method demonstrating its robustness and accuracy.