An efficient trajectory tracking algorithm for the backward semi-Lagrangian method of solving the guiding center problems

In this paper, we develop an effective numerical algorithm that tracks the trajectory needed to solve guiding center models using the backward semi-Lagrangian method. In terms of numerical calculations, two appreciably fast algorithms for the departure points are designed. One is a completely explicit formula for numerical solutions of the discrete system for each Cauchy problem. This formula is characterized by numerical factors that are less than half the multiplication number used in the usual Gaussian elimination. The other finds the required departure points with an interpolation method that is at least 30% less expensive for heavy Cauchy problems. Last, we propose a method to modify the solution to improve estimation of physical quantities, such as conservation of mass, which can be lost in interpolation solutions calculated at the departure points. It turns out that the proposed method not only saves a great deal of computation time, but also preserves physical quantities such as mass and total kinetic energy much better than conventional methods. To demonstrate the numerical evidence, we use the proposed method to simulate several problems such as the incompressible Euler equation, Kelvin-Helmholtz instability, Diocotron instability and a three-dimensional guiding center model.