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An analysis of the computational stability criteria for explicit and implicit integration schemes using a two layer model in Phillips [sigma] Coordinate
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    "The linear computational stability criterion for a multi-level primitive equation model is usually most strongly dependent upon the gravitational modes of oscillation. Recently, A. Robert and others have proposed the use of semi-implicit integration methods for reducing this dependence. In this note, we shall analyze the computational stability of three methods for integration of a simple fluid model which possesses only gravitational modes. The model will be written using the a coordinate proposed by Phillips and will consist of only two layers. Such a simple partitioning of the vertical structure gives rise to an especially compact problem but is of non-trivial interest. The three integration methods which are investigated are the explicit, the semi-implicit and a modified semi-implicit scheme. We arrive at a characteristic equation which is formally the same for all three methods. It is evaluated for an isothermal and standard atmosphere"--Introduction.
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