An iteration free backward semi-Lagrangian scheme for guiding center problems

Abstract

In this paper, we develop an iteration free backward semi-Lagrangian method for nonlinear guiding center models. We apply the fourth-order central difference scheme for the Poisson equation and employ the local cubic interpolation for the spatial discretization. A key problem in the time discretization is to find the characteristic curve arriving at each grid point which is the solution of a system of highly nonlinear ODEs with a self-consistency imposed by the Poisson equation. The proposed method is based on the error correction method recently developed by the authors. For the error correction method, we introduce a modified Euler's polygon and solve the induced asymptotically linear differential equation with the midpoint quadrature rule to get the error correction term. We prove that the proposed iteration free method has convergence order at least 3 in space and 2 in time in the sense of the L2-norm. In particular, it is shown that the proposed method has a good performance in computational cost together with better conservation properties in mass, the total kinetic energy, and the enstrophy compared to the conventional second-order methods. Numerical test results are presented to support the theoretical analysis and discuss the properties of the newly proposed scheme.