Dimensionally reduced models of fluid flow are derived from the full Euler equations by vertical averaging. These models include the hyperbolic shallow water model and the dispersive Green-Naghdi model. Usually these models are used for one layer of fluid. In [1] we have investigated the generalization of these models to two layers. The biggest difficulty associated with these two-layer models is in fact that they might not be well-posed everywhere. The Green-Naghdi model is unconditionally ill-posed while the shallow water model is for some values well-posed and for others ill-posed. We have shown how to regularize these models by fourth order dissipation. Numerical methods for solving both hyperbolic and dispersive models have been developed. The numerical codes for complicated Green-Naghdi models have been created by automatic code generation facilities. The effects of ill-posedness of the models and their regularization have been demonstrated on several examples. During the development of difference schemes for 1D hyperbolic shallow water equations [1] we have found an interesting idea of composite schemes consisting of several dispersive Lax-Wendroff steps followed by a diffusive Lax-Friedrichs step which serves as a consistent filter removing dispersive oscilations behind the shocks. The composite schemes work well also in higher dimensions and for other systems of conservation laws. We have used composite schemes for 2D shallow water simulations in the plane [2] and on the surface of a rotating sphere [3]. For the development and analysis of numerical methods we heavily use computer algebra packages which provide efficient and reliable tools for processing lengthy and tedious formulas~[4].

References

[2] Liska,R. - Wendroff,B.: Two-dimensional shallow water equations by composite schemes Int. J. Numer. Meth. Fluids, 30, 461--479, 1999.

[3] Liska,R. - Wendroff,B.: Shallow water conservation laws on a sphere Eighth International Conference on Hyperbolic Problems, Magdeburg, 2000.

[4] Liska,R. - Wendroff,B.: Where numerics can benefit from computer algebra in finite difference modelling of fluid flows In V. Ganzha, E.W. Mayr and E.V. Vorozhtsov, editors, Computer Algebra in Scientific Computing, CASC-99, 268-286, Berlin, Springer-Verlag, 1999.

Both authors were partially supported by the CHAMMP program of the US DOE. R. Liska was supported in part by the NSF grant CCR-9531828 and by the Ministry of Education of Czech Republic program Kontakt, Project ME 050 (1997-1999).

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