In this work, we study and extend a class of semi-Lagrangian exponential methods, which combine exponential time integration techniques, suitable for integrating stiff linear terms, with a semi-Lagrangian treatment of nonlinear advection terms. Partial differential equations involving both processes arise for instance in atmospheric circulation models. Through a truncation error analysis, we show that previously formulated semi-Lagrangian exponential schemes are limited to first-order accuracy due to the discretization of the linear term; we then formulate a new discretization leading to second-order accuracy. Also, a detailed stability study is conducted to compare several Eulerian and semi-Lagrangian exponential schemes, as well as a well-established semi-Lagrangian semi-implicit method, which is used in operational atmospheric models. Numerical simulations of the shallow-water equations on the rotating sphere are performed to assess the orders of convergence, stability properties, and computational cost of each method. The proposed second-order semi-Lagrangian exponential method was shown to be more stable and accurate than the previously formulated schemes of the same class at the expense of larger wall-clock times; however, the method is more stable and has a similar cost compared to the well-established semi-Lagrangian semi-implicit method; therefore, it is a competitive candidate for potential operational applications in atmospheric circulation modeling.

翻译：本文研究并拓展了一类半拉格朗日指数方法，该方法将适用于处理刚性线性项的指数时间积分技术与非线性平流项的半拉格朗日处理相结合。同时包含这两种过程的偏微分方程常见于大气环流模型。通过截断误差分析，我们发现先前构建的半拉格朗日指数格式因线性项离散化限制仅具有一阶精度；进而提出一种新的离散化方案以实现二阶精度。此外，本研究通过详细的稳定性分析，比较了若干欧拉与半拉格朗日指数格式，以及业务化大气模型中采用的成熟半拉格朗日半隐式方法。通过在旋转球面上对浅水方程进行数值模拟，评估了各方法的收敛阶数、稳定性特征及计算成本。结果表明：所提出的二阶半拉格朗日指数格式虽需更多计算时间，但比同类已有格式具有更好的稳定性和精度；与成熟的半拉格朗日半隐式方法相比，该方法稳定性更优且计算成本相当，因此在大气环流建模领域具有潜在的业务化应用前景。