We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.

我们提出了一种有效的半拉格朗日方法，用于在粗糙网格上高精度求解二维不可压缩的欧拉方程。这种新方法通过使用特征映射（CM）方法[1]和梯度增强水平集（GALS）方法[2]的组合来演化流程图。流图具有半组结构，可以将长时间的变形分解为短时间的子图。这导致了在线性时间中实现指数分辨率的数值方案。提供误差估计并分析保护特性。在涡旋合并，四模式和随机流动问题中说明了该方法的计算效率和高精度。还提出了与柯西-拉格朗日方法的比较[3]。