Volume 22, Issue 6

Elliptic boundary value problems for the Helmholtz equation are considered in the paper. In order to estimate their solutions and also the gradient of the solution, a special probabilistic representation and a corresponding statistical algorithm are constructed. The algorithm is a version of the process of ‘walking over spheres and in balls’ for solving a system of local integral equations obtained from the original differential problem by using the central and noncentral Green functions for the Helmholtz operator in a ball.

A projection (spectral) method is proposed and implemented for solution of a system of nonlinear partial differential equations describing a three-dimensional plasma wakefield. Comparison with results of modelling by a finite difference method is performed. A qualitative difference in the behavior of algorithms is indicated in calculation of functions with gradually decreasing smoothness.

A numerical method is proposed in this paper for studying hydrodynamic flows of heavy nonviscous fluid with a free surface over an arbitrary bed profile. This arbitrary bed profile is approximated by a piecewise-linear function splitting it into a finite number of domains with a step boundary. In order to implement this method, a quasi-two-layer model of fluid flow over a step boundary is used taking into account physical processes near the step. In this paper we implement numerical modelling of various physical phenomena, such as fluid flow over a sloping bed surface, shock wave creeping on a sloping bank, a fluid column fall over a bed surface given in various ways, fluid flow over a bed surface of a complex profile.

In the paper we present a new result for evaluating the convergence rate of iterative Newton-type methods with respect to the number of iteration steps. We obtain an explicit asymptotically correct estimate that provides a fruitful basis for many practical situations. As examples of such applications, we solve four important problems arising in numerical integration of ordinary differential equations and index 1 semi-explicit differential-algebraic systems.

An algorithm for the construction of a piecewise-linear outer hull of a three-dimensional object given by a set of points is presented in the paper. This algorithm may be applied to the initial stage of image recognition or initial triangulation of an object with subsequent refinement. The algorithm described here is finite. Examples of its application are presented.

A discrete maximum principle is presented for linear algebraic systems with coefficients of arbitrary signs and the number of unknowns exceeding (in the general case) the number of equations. The presentation uses the formal language of linear algebra (without any notions from the theory of difference schemes). This formal algebraic approach allows one to reveal easily the existence or absence of data for realization of an extended maximum principle in any algorithm represented in the form of a system of linear algebraic equations. The theorems formulated here directly imply the absolute stability of solutions to systems of equations. Variants of homogeneous algebraic systems and systems of equations with a right-hand side are considered. Theoretical results are illustrated by examples of actual problems.