Volume 27, Issue 6

A method for construction of piecewise-harmonic interpolations is proposed for spherical surfaces. Isometric coordinate transformations and piecewise-polynomial bases are used. The error of approximation by constructed interpolants is studied. The results of interpolation from the ARGO buoy system are presented.

Equations of a landslide motion over an uneven underwater slope subject to gravity and buoyancy forces, water friction and resistance are presented. A simulation of surface waves generated by a landslide in a bounded reservoir with a parabolic bottom profile has been performed within the nonlinear shallow water equations, and the results of that simulation are given. The influence of the parameters of the motion equation on the maximal splashing size is studied numerically.

In this paper, we consider the Neumann boundary value problem for the 3D diffusion equation with a diagonal diffusion tensor in a rectangular domain. The problem is discretized on rectangular meshes by the simplest version of the finite volume method. We assume that the coefficients of the discrete problem are dominant in the ‘horizontal’ xy -direction. A new approach based on earlier publications [5-7] for the construction of a multilevel preconditioner is proposed and analyzed. The main idea of the approach is to coarsen the discrete problem only in xy -direction, keeping the mesh in z -direction unchanged. We prove that the proposed preconditioner is spectrally equivalent to the system matrix and that the arithmetical complexity of the PCG method with this preconditioner is almost optimal. Numerical results are given.

We consider a class of convex variational problems and deduce computable and unconditional upper bounds of quantities, which are certain measures of errors associated with an approximation. Also, we discuss closely related mathematical questions, which are important for the a posteriori error estimation theory of nonlinear problems. Namely, we present generalized forms of the Prager- Synge estimate and of the Mikhlin’s variational identity, a generalized form of the Helmholtz decomposition theorem, and derive a general estimate of the distance to the set equilibrated fields.

An algorithm for numerical solution of Navier-Stokes equations for two-dimensional motion of a viscous heat-conducting gas is proposed in the paper. The discretization of the equations is performed by a combination of the method of trajectories for the substantive derivative and the finite element method with piecewise-bilinear basis functions for other summands. Results of numerical study of the supersonic flow structure are presented for a flat channel in its step expansion zone for a wide range of Mach and Reynolds numbers. Velocity and pressure fields are investigated, the vortex structure of the circulation flow is studied in the domain after the step

General problems in construction of adaptive grids and difference schemes on movable grids are discussed in the paper using the example of a predictor-corrector scheme for a linear transfer equation with variable coefficients, which is a model equation for testing calculation algorithms for two-dimensional incompressible fluid flows.