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.传统上，有限差分法（Finite difference methods）被用于数值天气预报（numerical weather prediction, NWP）的时域建模。时间谱解法（Time-spectral solution）凭借精度与效率双重优势，且可规避显式有限差分法中典型的类库朗（CFL）因果准则带来的时间步长限制，成为极具吸引力的替代方案。本研究采用时间谱解法——广义加权残数法（Generalized Weighted Residual Method，简称GWRM），对洛伦兹1984年混沌方程进行求解。研究对比了显式与隐式时间步进算法的精度与计算效率，结果表明GWRM的计算效率可与上述算法媲美，尤其在高精度场景下优势显著。针对微扰模拟场景，GWRM的运算速度最高可达有限差分法的四倍，核心原因在于其时间间隔通常比有限差分法大两个数量级。此外，GWRM还可生成切比雪夫级数展开（Chebyshev series expansions）形式的解析解。本研究结果令人鼓舞，为后续开展包含空间依赖性在内的、关于时间谱解法在数值天气预报建模中应用价值的相关研究提供了有力支撑。