A time-spectral approach to numerical weather prediction

Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal CFL-like criteria, typical for explicit finite difference methods, are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM (Generalized Weighted Residual Method). Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling.

有限差分方法在数值天气预报（NWP）中传统上用于模拟时间域。由于精度与效率的双重优势，以及避免了因果 CFL 类似标准（该标准通常与显式有限差分方法相关联的时间步长限制），时间-频谱解法成为了一种颇具吸引力的替代方案。在本研究中，我们采用广义加权残差法（GWRM）时间-频谱算法求解洛伦兹1984年的混沌方程。对显式和隐式时间步进算法的精度与效率进行了比较。研究发现，GWRM 的效率与这些方法相媲美，尤其是在高精度情况下。对于扰动场景，GWRM 的速度是有限差分方法的四倍之多。主要原因在于，GWRM 的时间间隔通常比有限差分方法大两个数量级。此外，GWRM 还具有将解析解表示为切比雪夫级数展开的优势。这些结果令人鼓舞，为进一步研究时间-频谱方法在 NWP 模型中的应用，包括空间依赖性研究，提供了依据。