摘要
本文研究是屬於數值天氣預報，或大氣理論模擬研究中，特殊數值方法之探討。文中著重Smolarkiewicz及Hsu-Arakawa正定義方法基本特性之討論。這些正定義數值方法可以保証平流計算之正（反）物理場，恆保持正（負）值。同時正定義計算方法可以有良好的質量保守，物理場向下游傳送，不受遠場影響之特性。我們以高斯波、正弦波以及方形波等不同數學特質之一維平流，探討正定義法的精確度（收斂速度）、極大值之衰減、均方根誤差、穩定度。測試正弦波、高斯波結果顯示Smolarkiewicz（1983）低階法（SML）的相位有明顯往上游方向偏移，而且SML其校正步驟的反覆次數至多兩次即可。Smolarkiewicz高階法（SMH）則產生有正確的相位，在兩次校正後的振幅強度衰減很小（繞一圈320時距後，維持原極大值之93%）。Hsu和Arakawa（1990）方法的相位也十分準確，振幅的衰減程度（約原極大值之86.5%）比SML經二次校正（約原極值之80%）為優。Smolarkiewicz混合階法（SMM）和SMH的結果幾乎相同。因SMM計算項數較少，所以我們認為SMM是可用於氣象預報模式高精確度、高效率的正定義方法。在方形波的平流探討中，Hsu-Arakawa及Smolarkiewicz法皆產生似合理之值，但此二方法都不具效率性。為比較起見，本文亦包括上游法，二階中差分法（FD2）及四階空間二階時間中差分法（FD4）之比較。
並列摘要
In this paper we give a detail comparative studies of two recent positive definite methods of Smolarkiewicz (1983, 1984) and Hsu and Arakawa (1990). We have tested the schemes with one dimensional linear advection. The Gaussian, Gaussian pus constant, sine and square waves are used as initial conditions. The convergent rate, stability, accuracy, maintenance of peak value, efficiency and the implementation of the methods are discussed. A new mixed order of Smolarkiewicz method is proposed. The mixed order scheme yields the same result as the high order of Smolarkiewicz scheme.Our computations indicate that no more than three corrective steps should be used in the mixed order and the high order of Smolarkiewicz schemes. Maximum of two corrective steps should be used in the low order scheme of Smolarkiewicz. The higher order scheme in the third corrective steps have the best peak value (almost 100% after 320 time steps). The high order scheme in the second corrective steps have 93% of the original peak value. The low order Smolarkiewicz in second corrective schemes have 80% and the Hsu-Arakawa scheme both have 86.5% of the original peak value. The low order scheme have upstream phase shift. The high, mixed order of Smolarkiewicz schemes and Hsu-Arakawa scheme do not have phase shift. The central fourth, second order finite differencing and upstream scheme are included for the comparison.
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