Volume 23, Issue 4

Some new operators appearing in problems of geophysical hydrodynamics are introduced. Properties of these operators are studied and estimates of their spectrum bounds are obtained. An application of the obtained results for the formulation of solution algorithms for the considered problems is presented.

A recently developed Cross 3D algorithm is applied to approximation of the electron density function. The algorithm is proved to be fast and reliable on a sample of quantum chemistry data produced by the MOLPRO package.

Unstable periodic trajectories of a chaotic dissipative system belong to the attractor of the system and are its important characteristics. Many chaotic systems have an infinite number of periodic solutions forming the skeleton of the system attractor. This allows one to approximate the system trajectories and statistical characteristics by using periodic solutions. The least unstable orbits may generate local maxima of the system state distribution functions on the attractor. With respect to atmospheric systems this means that orbits may determine dynamic circulation regimes and typical variability modes of the system. In some cases, given a small number of periodic solutions, one can describe the dynamics on the attractor of the system and the basic statistics with sufficient precision. Thus, the information concerning periodic trajectories of a particular dynamic system may be very important for analysis of its behavior. A search for periodic trajectories is reduced to the solution of a system of nonlinear equations with respect to the initial condition of an orbit and its period. The choice of a numerical solution method and an initial guess is an important aspect here. In this paper we consider the problem of the calculation of periodic trajectories for a barotropic model of the atmosphere. Several methods for determination of periodic orbits of the model are formulated and implemented, including the Newton method with step suppression, the Newton method with a second order tensor correction, the quasi-Newton method with step suppression, the quasi-Newton method with minimization of the error functional and approximate Hessian inversion by the LBFG scheme and the GMRES method. A comparison of the efficiency of these numerical methods and different choices of initial conditions is performed. Various factors influencing the rate of convergence of the methods are considered.

Methods of iterative solution in Krylov subspaces are considered for systems of linear algebraic equations with positive definite nonsymmetric real matrices A , where directing (correcting) vectors and residual vectors possess different properties of A -semiconjugacy or orthogonality. Variational and projective properties of some variants of such algorithms and the possibility to use dynamic preconditioners are described. Results of numerical experiments are presented.

In the paper a splitting method used in practice for numerical solution of primitive equations is justified. The idea of justification is similar to the one proposed by R. Rannacher with respect to the Chorin scheme for the Navier–Stokes equations. It is suggested that a solution to the splitting scheme tends to a solution to the primitive equations as the time step tends to zero.

We consider the following problem for an operator S acting in a Hilbert space H and specifying for any u 0 ∈ H an evolutionary process u i +1 = S ( u i ), i = 0, 1,…: for given points z 0 , a 0 construct corrections f i ∈ ℱ from a fixed subset ℱ ⊂ H so that positive semitrajectories and approach each other, i.e. || a i – z i || 2 → 0 for i → ∞. Methods for construction of desired corrections f i are proposed under some restrictions, and results of numerical calculations are presented for the one-dimensional Chafee–Infante equation.