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.

在本文中，我们开发了一种有效的数值算法，该算法使用后向半拉格朗日方法来跟踪求解制导中心模型所需的轨迹。在数值计算方面，设计了两种出发点的快速算法。一个是针对每个柯西问题的离散系统数值解的完全明确的公式。该公式的特征在于数值因子小于通常的高斯消除所使用的乘法数的一半。另一个使用插值方法找到所需的出发点，对于严重的柯西问题，该方法至少便宜30％。最后，我们提出一种修改解决方案的方法，以改善对物理量的估算，例如质量守恒，在出发点计算的插值解中可能会丢失。结果表明，与传统方法相比，该方法不仅节省了大量的计算时间，而且还保留了质量，总动能等物理量。为了证明数值证据，我们使用提出的方法模拟了一些问题，例如不可压缩的Euler方程，Kelvin-Helmholtz不稳定性，Diocotron不稳定性和三维制导中心模型。