Band 5, Heft 3

Unstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

A numerical algorithm for solving the Navier-Stokes equations for incom- pressible viscous fluid in an arbitrary two-dimensional region on nonstaggered grids is presented. The idea of the transition to a general curvilinear coordinate system, trans- forming the physical region into a parametrical square is used. For the discrete solution an unconditional a priory estimate has been obtained. The results of the benchmark computations for a driven skewed cavity flow and the results of the fluid flow modeling in a cavity of an arbitrary shape are given.

In this work we propose a new method for solving linear elliptic equations in an unbounded domain. The method is based on the representation of the exact solution as the sum of two functions. The former is the solution of some auxiliary problem and the latter can be found using the Green formula. Using finite-difference schemes, this method has a quadratic order of accuracy independent of the size of the computational domain, and in the 2D case requires O(N³) operations to find the solution, where N³ is the number of nodes within the computational domain. In the 3D case the method requires O(N^4) operations. Test computational examples showing the method's efficiency are given.

This paper presents Iterative Scalable Smoothing (ISS), a new itera- tive multi-scale method for multivariate interpolation of scattered data. Each iteration step in the process reduces the residues of the current interpolation result by appli- cation of a smoothing operator to a piecewise constant function that interpolates the residues of the current interpolant, and by adding the resulting function to the current approximation, which is initially set to zero. The convergence of the method is proved and conditions for the di®erentiability of the convergence result are given. For a uni- form mesh an e±cient algorithm is constructed, for which the numerical complexity is estimated. Several 2D numerical examples illustrate the theoretical results. By a 3D test with several test-functions and random nodes it is shown that the accuracy of the proposed method is comparable with the quadratic modiffcation of Shepard's method, which is known to be more accurate than triangle-based methods. Then, in 1D tests, by stochastic simulation with random nodes and random functions the ISS method is compared with the Cubic Splines method, Shepard's method and the Kriging method. We also compare the stability of these methods with respect to noisy data. For the special case of regular nodes, properties of the method are verified by comparing its 1D response function with the response function of the cubic spline and the perfect interpolator (the sinc-function). Special attention is paid to the e®ect of the tuning parameter

Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using gen- eral irregular grids. The vector analysis di®erence operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the di®erence operators constructed on two types of grids are investigated. Construction and analysis of the di®erence schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-diffusion problems. The other examples concerned the solution of the non- stationary vector problems described by the second-order equations or the systems of first-order equations.