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 convergence 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. Rather than simply using four iterations of a second order Crank-Nicolson time discretisation as is customary, the present article studies the impact of using various third-order, four stage Rosenbrock-Wanner schemes, where instead of a simple time centering, the integration weights are chosen to meet specific stability and order conditions. 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, a desirable property in the presence of fast wave dynamics across multiple scales. These schemes are compared to four iterations of a Crank-Nicolson scheme for the solution of the 2D rotating shallow water equations at the 3D compressible Euler equations at both planetary and non-hydrostatic scales are are shown to exhibit improved results in terms of their energetic profiles and stability.

翻译：非静力大气模型常采用半隐式时间离散格式，以规避显式处理快速声波和重力波时的时间步长限制。使用牛顿法迭代求解该方程组至收敛被认为计算成本过高，因此非线性求解器通常被截断为固定迭代次数，并采用每时间步仅重构一次的近似雅可比矩阵。本文并未沿用常规的二次Crank-Nicolson时间离散格式的四次迭代方案，而是系统研究了采用多种三阶四阶段Rosenbrock-Wanner格式的影响——这些格式通过选取满足特定稳定性与阶数条件的积分权重，取代了简单的时间中心化处理。Rosenbrock-Wanner格式因其在仅使用近似雅可比矩阵时仍能保持时间阶数的特性而展现出良好前景，且可构造为刚性稳定格式，这对于涉及多尺度快波动力学的系统而言是理想特性。研究将上述格式与四次迭代的Crank-Nicolson格式进行对比，分别求解行星尺度与非静力尺度下的二维旋转浅水方程组及三维可压缩欧拉方程组，结果表明：Rosenbrock-Wanner格式在能量特征与稳定性方面均展现出更优性能。