We conduct a thorough study of different forms of horizontally explicit and vertically implicit (HEVI) time-integration strategies for the compressible Euler equations on spherical domains typical of nonhydrostatic global atmospheric applications. We compare the computational time and complexity of two nonlinear variants (NHEVI-GMRES and NHEVI-LU) and a linear variant (LHEVI). We report on the performance of these three variants for a number of additive Runge-Kutta Methods ranging in order of accuracy from second through fifth, and confirm the expected order of accuracy of the HEVI methods for each time-integrator. To gauge the maximum usable time-step of each HEVI method, we run simulations of a nonhydrostatic baroclinic instability for 100 days and then use this time-step to compare the time-to-solution of each method. The results show that NHEVI-LU is 2x faster than NHEVI-GMRES, and LHEVI is 5x faster than NHEVI-LU, for the idealized cases tested. The baroclinic instability and inertia-gravity wave simulations indicate that the optimal choice of time-integrator is LHEVI with either second or third order schemes, as both schemes yield similar time to solution and relative L2 error at their maximum usable time-steps. In the future, we will report on whether these results hold for more complex problems using, e.g., real atmospheric data and/or a higher model top typical of space weather applications.

翻译：我们对适用于非静力全球大气应用中典型球面域的可压缩欧拉方程的不同形式的水平显式与垂直隐式（HEVI）时间积分策略进行了全面研究。比较了两种非线性变体（NHEVI-GMRES和NHEVI-LU）与一种线性变体（LHEVI）的计算时间与复杂度。针对从二阶到五阶精度的多种加性龙格-库塔方法，报告了这三种变体的性能表现，并验证了每个时间积分器下HEVI方法预期的精度阶数。为评估每种HEVI方法的最大可用时间步长，我们对非静力斜压不稳定现象进行了为期100天的模拟，并利用该时间步长比较了各方法达到解所需的时间。结果表明，在测试的理想化案例中，NHEVI-LU的速度是NHEVI-GMRES的2倍，而LHEVI的速度是NHEVI-LU的5倍。斜压不稳定与惯性重力波模拟指出，采用二阶或三阶方案的LHEVI是最优时间积分器选择，因为这两种方案在最大可用时间步长下具有相似的求解时间和相对L2误差。未来我们将报告这些结果是否适用于更复杂的问题，例如使用真实大气数据和/或空间天气应用中典型的更高模式顶高度的情况。