多守恒性的差分格式的分析和比较

Analysis and comparison of multi-conservation difference schemes

球面大气浅水波方程(在忽略摩擦和外源强迫的条件下)具有总能量、总质量、总位涡、总位涡拟能和总角动量守恒等五个重要的物理守恒性。然而,将大气浅水波方程差分离散化后,常常不能保持这些守恒性。本文对新发展的两种较好的差分格式进行比较;一种可称之为修正总能量守恒格式,它可以较好地保持上述五个守恒性中的四个;而另一种可称之为准辛格式,它可以使上述五种守恒性均近似保持。文中对这两类格式作了具体分析,并作相应的数值试验和比较,结果表明这两类格式均值得推广应用。

The spherical shallow water equations without the effects of any outer frictions and forcing source have many remarkable features, such as the energy conservation, the mass conservation, the potential vorticity conservation, the potential vorticity enstrophy conservation and the absolute angular momentum conservation. Once the equations are discretized, however, these important features can hardly be maintained. Focusing on the multi-conservation problem of numerical method, two new difference schemes developed recently are discussed and compared in this paper. One is the improved energy-conservation scheme, by which four of the aforesaid conservation properties can be kept well. The other is the symplectic-like scheme, which can conserve all of the five physical integrals approximately. Numerical tests show that both of the schemes are with good multi-conservation features and worth generalizing and applying.