For simulations of time-evolution problems, such as weather and climate models, taking the largest stable time-step is advantageous for reducing the wall-clock time. We propose methods for studying the effect of linear dispersive errors on the time-stepping accuracy of nonlinear problems. We demonstrate an application of this to the Rotating Shallow Water Equations (RSWEs). To begin, a nonlinear time-stepping `triadic error' metric is constructed from three-wave interactions. Stability polynomials, obtained from the oscillatory Dahlquist test equation, enable the computation of triadic errors for different time-steppers; we compare five classical schemes. We next provide test cases comparing different time-step sizes within a numerical model. The first case is of a reforming Gaussian height perturbation. This contains a nonlinear phase shift that can be missed with a large time-step. The second set of test cases initialise individual waves to allow specific triads to form. The presence of a slow transition from linear to nonlinear dynamics creates a good venue for testing how the slow phase information is replicated with a large time-step. Three models, including the finite element code Gusto, and the MetOffice's new LFRic model, are examined in these test cases with different time-steppers.

翻译：针对时间演化问题的模拟（如天气与气候模型），采用最大稳定时间步长有利于缩短计算耗时。我们提出研究线性色散误差对非线性问题时间步进精度影响的方法，并以旋转浅水方程（RSWEs）为例进行应用演示。首先，基于三波相互作用构建非线性时间步进"三元误差"指标；通过振荡型Dahlquist测试方程获取稳定性多项式，进而计算不同时间步进方案的三元误差，并比较五种经典格式。随后，我们提供数值模型中比较不同时间步长效果的测试案例。首个案例为高斯高度扰动重构过程，其中包含非线性相移，该现象在使用大步长时可能被遗漏。第二组案例通过初始化单波以形成特定三波组，从线性到非线性动力学的缓慢过渡过程为验证大步长下慢相信息复现效果提供了良好场景。本研究在三种模型中（包括有限元代码Gusto及英国气象局新型LFRic模式）结合不同时间步进格式进行测试。