One-step L(α)-stable temporal integration for the backward semi-Lagrangian scheme and its application in guiding center problems

We present a backward semi-Lagrangian method with third-order temporal accuracy for solving the guiding center problem. For solving highly nonlinear characteristic curves, we propose an iteration-free L(α)-stable method equipped with an error correction technique and a classical collocation method. As a discretization of the Poisson equation, we adopt the fourth-order centered differential scheme and the Shortley–Weller discretization depending on the boundary of the domain, and apply the local cubic interpolation polynomial for the evaluation of the density and the potential at the non-grid foot points of the characteristics. The novelty of the proposed method is its good stability as well as its outstanding computational cost compared to those of the recently developed high-order Adams–Moulton method. Furthermore, based on several numerical tests, we conclude that the proposed method exhibits excellent performance for long-time stable simulations and allows for large temporal step sizes.