Vertically Averaged Stratified Fluid Flow Models - Analysis and Numerical Solution Supported by Computer Algebra Tools ------------------------------------------------------------------- by Richard Liska+ and Burton Wendroff* + Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Brehova 7 115 19 Prague 1 Czech Republic liska@siduri.fjfi.cvut.cz http://www-troja.fjfi.cvut.cz/~liska *Group T-7, Los Alamos National Laboratory Los Alamos NM 87544 USA Vertical averaging, used often in ocean and atmosphere modelling, reduces the dimension of the Euler equations, describing non-viscous, incompressible fluid flow, and results in vertically averaged models which do not have vertical coordinate and thus can be solved much faster than full-dimensional models. There are two well known one-layer vertically averaged models, namely shallow water and Green-Naghdi models. Here we deal with two-layer extensions of these models. As the complexity of these models is high (especially the Green-Naghdi one), the use of computer algebra tools during the analysis and numerical code development has been essential. Well-posedness of the models has been checked by the Fourier method where in one case a special quantifier elimination algorithm has been employed to obtain final conditions of ill-posedness. For shallow water models, the composition of Lax-Wendroff and Lax-Friedrichs difference schemes with good shock resolving capabilities has been used. The ill-posed Green-Naghdi models have been treated by an implicit backward Euler difference scheme. Code generation of an algorithm solving this non-linear scheme by full Newton method resulted in a numerical source code of the size about 1/2 MB. Without computer algebra tools, the treatment of these models would not be possible.