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Baroclinic stability for a family of two-level, semi-implicit numerical methods for the 3D shallow water equations. (English) Zbl 1287.76172

Summary: The baroclinic stability of a family of two time-level, semi-implicit schemes for the 3D hydrostatic, Boussinesq Navier-Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of V. Casulli and R. T. Cheng [Int. J. Numer. Methods Fluids 15, No. 6, 629–648 (1992; Zbl 0762.76068)], is examined in a simple 2D horizontal-vertical domain. It is demonstrated that existing mass-conservative low-dissipation semi-implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two-step predictor-corrector schemes is proposed that have better theoretical characteristics than existing single-step schemes.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
76E20 Stability and instability of geophysical and astrophysical flows
86A05 Hydrology, hydrography, oceanography

Citations:

Zbl 0762.76068

Software:

TRIM
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References:

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