Finite-Di erence Approximations for Cosymmetry Preservation in a Filtration Convection Problem

Finite-diierence approximations preserving the cosymmetry property for two-dimensional ltration convection problem are presented. Cosymmetry preserving discretizations connrm the predicted behavior of the family of equilibria and degeneration of the family is observed when an inappropriate approximation was used. 1 Planar Darcy convection problem and cosymmetry In recent years several discretization methods are developed to preserve the qualitative properties of the underlying continuous dynamical systems. In this work we study the conservation of cosymmetry which was introduced by Yudovich 3]. A number of interesting phenomena were found for both ode's and pde's possessing the cosymmetry property. Particularly, the cosymme-try may be a reason for the existence of continuous family of equilibria with a stability spectrum depending on the location of a point on the family. As a result, the family may be formed by stable and unstable regimes. Cosymmetry is in a sense dual to the symmetry notion. It is also related to the more general concept of implicit constraints for dynamical systems 5] and to conservation laws in classical mechanics 6]. A number of interesting eeects were found in the planar ltration-convection problem of uid ow through porous media 2], 3], 4]. A Galerkin method was used for spatial discretization and the resulting nite-dimensional systems of small size were studied in 2]. In this note using a cosymmetry preserving nite-diierence approximation, systems of larger size are investigated. We consider the two-dimensional ltration convection of a saturated incompressible viscous uid in a rectangular container D = 0; a] 0; b] lled with porous medium, where the thermal convection equations are described by Darcy law. The temperature diierence T was held constant between the lower y = 0 and upper y = b boundaries of the rectangle and the temperature on the vertical boundaries obeys a linear law, so that a time independent uniform vertical temperature proole is formed. We consider perturbation of the velocity (here in form of the stream function (x; y; t)) and temperature (x; y; t) from the basic state of rest with a linear conductive proole. The dimensionless equations of ltration convection problem are 3]: (2) where J(f; g) denotes the Jacobian operator, is the Rayleigh number. For a given initial temperature distribution 0 , the stream function can be obtained from (1), (2) as solution of the Dirichlet problem via Green's operator = GG x , which is the cosymmetry for the system (1)-(2) 4].