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 first 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 a second-order accurate method. Also, a detailed stability study, both considering a linear stability analysis and an empirical simulation-based one, 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, considering standard and challenging benchmark test cases, 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; 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，该类方法结合了适用于刚性线性项积分的指数时间推进技术与非线性平流项的半拉格朗日处理方式。在诸如大气环流模型等场景中，会同时出现包含这两种过程的偏微分方程。通过截断误差分析，我们首先揭示了先前提出的半拉格朗日指数格式因线性项离散化而仅限于一阶精度；继而提出一种新的离散化方案，从而实现了二阶精度方法。此外，我们进行了详细的稳定性研究——既包括线性稳定性分析，也包含基于模拟的经验性评估——以比较多种欧拉及半拉格朗日指数格式，同时对比了业务化大气模型中常用的半拉格朗日半隐式方法。通过旋转球面浅水方程的数值模拟，采用标准及具有挑战性的基准测试案例，评估了各方法的收敛阶、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方法相比同类先前格式具有更高的稳定性和精度，但需以更大的壁钟时间为代价；然而，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的实际业务应用中具有良好的竞争潜力。