大气动力学方程组的半拉格朗日计算方案的数学研究
MATHEMATICAL STUDY ABOUT SEMI-LAGRANGIAN SCHEME IN ATMOSPHERIC DYNAMICS EQUATIONS
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摘要: 通过对大气动力学方程组的半拉格朗日方案的数学分析,得出主要结论如下:(1)说明了Robert的半拉格朗日方案并不是绝对稳定的,通过特征线理论和双曲拟线性方程组解的理论,进一步说明了Robert理论的不正确。(2)通过分析右端项沿轨道积分,给出了一个关于半拉格朗日方案成立的判据,该判据与CLF相仿。(3)根据浅水方程特征理论,发现半拉格朗日方案应包括沿轨道反向积分(大气中慢过程)和沿特征锥积分(大气中快过程),而已有的方案仅含前一类。因此,今后有必要研究后一类的半拉格朗日方案并研究这两类过程之间的相互作用的计算问题。(4)即使研究沿轨道反向积分问题,其特征跟原方程组解直接有关,因此,不仅仅是一个常微分方程组的问题。而大气动力学方程组的经典解一般仅在小范围成立,并且一维、二维和三维间断均会出现,故大范围反向积分特征线一般是不可能的。Abstract: By examining Semi-Lagrangian scheme in atmospheric dynamics,the main conclusions are obtained as follows.(1)The Semi-Lagrangian scheme by Robet is not of absolute stability.(2)The sufficient condition for computationally stability about Semi-Lagrangian scheme is obtained,which is similar with CFL condition.(3)According to characteristics of shallow-water equations,there are two kinds of characteristic surfaces:the stream lines(for slow process) and Monge cone (for fast process).However,the Semi-Lagrangian scheme only contains the first kind.Thus,the second kind and interaction between these kinds should be studied in the future.(4)As global classic solution in atmospheric dynamics equations does not exist generally,global backward integration is not carried out generally.
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