Filtration-Convection Problem: Spectral-Difference Method and Preservation of Cosymmetry Olga Kantur, Vyacheslav Tsybulin Department of Mathematics and Mechanics, Rostov State University, 344090 Rostov on Don, Russia kantur rsu@mail.ru, tsybulin@math.rsu.ru Abstract. For the problem of filtration of viscous fluid in porous medium it was observed that a number of one-parameter families of convective states with the spectrum, which varies along the family. It was shown by V. Yudovich that these families cannot be an orbit of an operation of any symmetry group and as a result the theory of cosymmetry was derived. The combined spectral and finite-difference approach to the pla- nar problem of filtration-convection in porous media with Darcy law is described. The special approximation of nonlinear terms is derived to preserve cosymmetry. The computation of stationary regime transfor- mations is carried out when filtration Rayleigh number varies. 1 Introduction In this work we study the conservation of cosymmetry in finite-dimensional mod- els of filtration-convection problem derived via combined spectral and finite- difference method. Cosymmetry concept was introduced by Yudovich [1,2] and some interesting phenomena were found for both dynamical systems possessing the cosymmetry property. Particularly, it was shown that cosymmetry may be a reason for the existence of the continuous family of regimes of the same type. If a symmetry group produces a continuous family of identical regimes then it im- plies the identical spectrum for all points on the family. The stability spectrum for the cosymmetric system depends on the location of a point, and the family may be formed by stable and unstable regimes. Following [1], a cosymmetry for a differential equation ˙ u = F (u) in a Hilbert space is the operator L(u) which is orthogonal to F at each point of the phase space i.e (F (u),L(u)) = 0,u ∈ R n with an inner product (·, ·). If the equilibrium u 0 is noncosymmetric, i.e. F (u 0 ) = 0 and L(u 0 ) = 0 , then u 0 belongs to a one-parameter family of equilibria. This takes place if there are no additional degeneracies. A number of interesting effects were found in the planar filtration-convection problem of fluid flow through porous media [1–4]. The investigations in [4] were carried out for finite-dimensional approximations of small size, so it is desirable to P.M.A. Sloot et al. (Eds.): ICCS 2002, LNCS 2330, pp. 432-441, 2002.  Springer-Verlag Berlin Heidelberg 2002