Non-hydrostatic atmospheric models often use semi-implicit temporal discretisations in order to negate the time step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the non-linear solver is typically truncated to a fixed number of iterations, using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third-order, four stage Rosenbrock-Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank-Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock-Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly-stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow water equations and the 3D compressible Euler equations at both planetary and non-hydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank-Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock-Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at non-hydrostatic scales.

翻译：非静力大气模型常采用半隐式时间离散化方法，以避免显式解析快速声波和重力波所需的时间步长限制。使用牛顿法将所得系统求解至机器精度被认为计算成本过高，因此非线性求解器通常截断至固定迭代次数，并使用每时间步仅重构一次的近似雅可比矩阵。本文研究了采用多种三阶四阶段Rosenbrock-Wanner格式（其积分权重根据特定稳定性和阶次条件选取）的影响，并与英国气象局LFRic模型中使用的Crank-Nicolson时间离散化方法进行了对比。Rosenbrock-Wanner格式因其仅需近似雅可比矩阵即可保持时间阶次，且可构造为刚性稳定以确保快速未解析模式衰减，成为具有前景的替代方案。本文针对二维旋转浅水方程和三维可压缩欧拉方程，在行星尺度与静力尺度下对这些格式进行了比较，结果表明其能量分布特性和稳定性均有改善。计算性能方面的结果则表现不一：Crank-Nicolson方法在行星尺度斜压不稳定性测试中允许更长的时间步长和更快的求解速度，而Rosenbrock-Wanner方法在非静力尺度上升气泡测试中允许更长的时间步长和更快的求解速度。