Smolarkiewicz正定義數值方法中的交錯項

The Spatial Cross Derivative Terms in the Smolarkiewicz Positive Definite Scheme

柳再明(Tzay-Ming Leou) ； 郭鴻基(Hung-Chi Kuo)

《大氣科學》 22卷2期 (1994/06) Pp. 277-296

摘要

Smolarkiewicz指出一數值平流方法應納入交錯項而未納入者是一不穩定法。我們針對Smolarkiewicz法交錯項加入與否、圓錐初始位置的改變、穩定度的探討以及邊界的影響進行多方面的探討。研究結果指出有納入交錯項的Smolarkiewicz法如Smolarkiewicz所言是一穩定的方法，而對於Smolarkiewicz法沒有加入交錯項的部份，只要注意△t大小的使用及圓錐所在位置，雖然Smolarkiewicz法沒有納入交錯項，也會是一穩定的方法。以分離法的執行多維運算方式，除了可以使用比直接二維運算較大的△t外，亦自動包含交錯項的效應，可以免除二維處理時，考慮交錯項是否納入的困擾。以上這些最重要的結果，總結於圖13至圖15，這是Smolarkiewicz法所沒有的重要補充，此外我們亦更正了Smolarkiewicz法多維公式之錯誤。

關鍵字

交錯項 ； 正定義數值方法

並列摘要

Smolarkiewicz pointed out that the multi-dimensional numerical advection scheme without the spatial cross derivative terms is an unstable scheme. We explore the impact of the spatial cross derivative terms in the Smolarkiewicz method. Both the analytical Taylor expansion and numerical calculations in two dimension are used in the study. Our results indicate that the Smolarkiewicz method without the cross terms can be a stable scheme if not too large △t is used or the position of the advected cone is not too near the computational boundary. Splitting process allow the usage of a large △t than the multi-dimensional calculations with the cross terms. In addition, the splitting process involves the cross spatial derivative terms automatically in the calculation. The main results of our work are summarized in fig. 13, 14 and 15, which complement the Smolarkiewicz results of 1983 and 1985. We also give the correct multi-dimensional cross spatial derivative terms for the Smolarkiewicz method.

並列關鍵字

cross derivative terms ； positive definite scheme