Volume 24, Issue 3

The temporal stability of Poiseuille flows in ducts of square cross-sections has been numerically studied. A new classification of disturbances based on their symmetries has been proposed. Stability characteristics such as location of leading eigenvalues, shape of eigenmodes, values of maximal energy growth, and critical Reynolds numbers have been computed.

We have developed a new monotone cell-centered finite volume method for the discretization of diffusion equations on conformal polyhedral meshes. The proposed method is based on a nonlinear two-point flux approximation. For problems with smooth diffusion tensors and Dirichlet boundary conditions the method is interpolation-free. An adaptive interpolation is applied on faces where diffusion tensor jumps or Neumann boundary conditions are imposed. The interpolation is based on physical relationships such as continuity of the diffusion flux. The second-order convergence rate is verified with numerical experiments.

A finite-difference representation describing the Coriolis force in numerical methods of the Godunov type for flows of rotating shallow water is proposed in the paper. Finite-difference schemes are proposed for modelling flows both on a flat bed and on a bed of an arbitrary profile. The influence of the Coriolis force is simulated by introducing a fictitious unsteady boundary. The numerical algorithms developed here are based on representing an arbitrary bed surface and the Coriolis force by a complex unsteady step boundary. Due to inhomogeneity of the bed surface and the influence of the Coriolis force, a quasi-two-layer model of a liquid flow over a step boundary is used for numerical approximation of the source terms. The model takes into account the hydrodynamic peculiarities. This allows us to give a visual interpretation of nonlinear processes caused by inhomogeneity of the bottom and by the fictitious unsteady boundary describing the Coriolis force. The use of the quasi-two-layer model provides a better approximation to the original Euler equations in the presence of source terms. Comparative analysis of the well-known finite-difference schemes describing the rotation and inhomogeneity of the bottom surface profile is performed. Numerical calculations confirming the efficiency of the proposed method are also performed.

Various weighted algorithms of numerical statistical modelling with trajectory branching are formulated and studied in the case where the next weight factor exceeds one. As the result, the ‘weight’ of a separate ‘branch’ does not exceed one and the variance of the estimate of the calculated functional is finite. The problems of unbiasedness and variance finiteness are solved based on the method of recurrent ‘partial’ averaging presented in this paper. As an application, estimates of the particle reproduction rate and the solutions to the Helmholtz equation are considered. The comparative computational cost of the method of ‘exponential transformation’ with branching is studied for the radiation transfer theory problems based on the Galton–Watson majorant process. In this connection, an expression for the second moment of the total number of particles in the Galton–Watson subcritical process is obtained. The minimum variance problem for an integer-valued random variable under a fixed mathematical expectation is also solved. In addition, variances of weighted algorithms with branching are considered for solution of integral equations with power nonlinearity.

In the paper we present several numerical algorithms to simulate the fractional Brownian motion (the generalized Wiener process). The algorithms have been developed on the basis of the integral representation of the fractional Brownian motion and the spectral decomposition for its increments. The convergence of the numerical methods has been studied.

A special basis based on trigonometric functions is proposed for solution of integral equations with kernels of the form K ( x – t ) by the Galerkin method. This basis possesses high approximation quality and allows one to reduce the double integral in the Galerkin algorithm to a very simple single integration.