Comparison of dimensionally split and multi-dimensional atmospheric transport schemes for long time steps
Comparison of dimensionally split and multi-dimensional atmospheric transport schemes for long time steps
复制标题
长时间步长的维度分割和多维大气传输方案的比较
DOI：
10.1002/qj.3125
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发表时间：
2017
影响因子：
8.9
通讯作者：
Chen Y
中科院分区：
文献类型：
--
作者：
Chen Y
分裂平流方案由于其在每个空间维度上的效率和精度而在大气模拟中具有吸引力。使用通量形式的半拉格朗日技术可以实现精确的长时间步长，而无需显著的成本。本文中使用的维度分裂格式是从一维分段抛物方法构造的，并使用COSMIC分裂扩展到二维。将分裂格式与真正的多维线法格式进行了比较，后者采用隐式时间推进，在柯朗数明显大于1时是稳定的。提出了笛卡尔平面上的二维平流试验，避免了球面域或多面元网格的复杂性。这些是固体旋转，地形上的水平平流和变形流。测试用例使用扭曲的非正交网格来表示倾斜地形或模拟立方体球体边缘附近的扭曲。网格扭曲预计会加剧与维度分裂相关的误差，但维度分裂方案的精度在网格扭曲的情况下只会略有下降。当使用长时间步长时，维度分裂方案也失去了一些精度。多维格式对网格畸变几乎完全不敏感，并且在高分辨率下渐近于二阶精度。正如隐式时间步进所预期的那样，当使用长时间步长时会出现相位误差，但空间分辨率良好的特征会以正确的速度平流，并且多维格式总是稳定的。计算成本（乘法次数）的简单估计表明，隐式格式是最昂贵的，特别是对于大的Courant数。如果使用具有显式时间步进的多维格式，则柯朗数被限制为小于1，精度保持不变，并且成本变得与维度分裂格式相似。
Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time steps can be achieved without significant cost using the flux‐form semi‐Lagrangian technique. The dimensionally split scheme used in this paper is constructed from the one‐dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi‐dimensional, method‐of‐lines scheme which, with implicit time‐stepping, is stable for Courant numbers significantly larger than 1.Two‐dimensional advection test cases on Cartesian planes are proposed which avoid the complexities of a spherical domain or multi‐panel meshes. These are solid‐body rotation, horizontal advection over orography and deformational flow. The test cases use distorted non‐orthogonal meshes either to represent sloping terrain or to mimic the distortions near cubed‐sphere edges.Mesh distortions are expected to accentuate the errors associated with dimension splitting, however the accuracy of the dimensionally split scheme decreases only a little in the presence of mesh distortions. The dimensionally split scheme also loses some accuracy when long time steps are used. The multi‐dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second‐order accuracy at high resolution. As is expected for implicit time‐stepping, phase errors occur when using long time steps but the spatially well‐resolved features are advected at the correct speed and the multi‐dimensional scheme is always stable.A naive estimate of computational cost (number of multiplies) reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi‐dimensional scheme is used instead with explicit time‐stepping, the Courant number is restricted to less than 1, the accuracy is maintained, and the cost becomes similar to the dimensionally split scheme.