卷 64, 编号 7 (2024)

In the approximate solution of boundary value problems for partial differential equations, difference methods are widely used. Grid approximations are most simply constructed when dividing the calculated area into rectangular cells. Usually the grid nodes coincide with the vertices of the cells. In addition to such nodal approximations, grids with nodes in the centers of cells are also used. It is convenient to formulate boundary value problems in terms of invariant operators of vector (tensor) analysis, which are compared with the corresponding grid analogues. The paper builds analogues of gradient and divergence operators on non-standard rectangular grids, the nodes of which consist of both the vertices of the calculated cells and their centers. The proposed approach is illustrated by approximations of the boundary value problem for the stationary two-dimensional convection-diffusion equation. The key features of the construction of approximations for vector problems with orientation to applied problems of solid mechanics are noted.

When developing numerical methods for solving nonlinear minimax problems, the following auxiliary problem arose: in the convex hull of some finite set in Euclidean space, find the point with the lowest norm. In 1971, B. Mitchell, V. Demyanov and V. Malozemov proposed a non-standard algorithm for solving this problem, which later became known as the MDM algorithm (based on the capital letters of the authors’ surnames). This article discusses a specific minimax problem: to find a ball of the smallest volume containing a given finite set of points. It is called the Sylvester problem and is a special case of the Chebyshev center of the set problem. The convex quadratic programming problem with simplex constraints is compared to the Sylvester problem. To solve this problem, the article suggests using a variant of the MDM algorithm. With its help, a minimizing sequence of plans is built, such that only two components differ from neighboring plans. The numbers of these components are selected based on certain optimality conditions. The weak convergence of the obtained sequence of plans is proved, from which follows the norm convergence of the corresponding sequence of vectors to the only solution of the Sylvester problem. Four typical examples on the plane are given.

The paper studies difference approximations of the optimal control problem with boundary observation of the conormal derivative of the state described by the Dirichlet problem for semi-linear elliptic equations with controls in the coefficients of the convective transport operator and the nonlinear term of the equation. The issues of the correctness of the formulation of the optimal control problem were considered. Differential approximations of the optimal control problem are constructed. The issues of convergence of approximations in terms of functionality and control are studied. The regularization of approximations is carried out.

The basics of fast trigonometric interpolation are given for nonperiodic functions, making it possible to obtain an approximate solution with high accuracy. Formulas are obtained to calculate the coordinates of the launch point of an aircraft with high accuracy due to the simultaneous application of the method for universal fast trigonometric interpolations and the method for extrapolations at the ends of a certain specified segment. Numerical experiments show that after the launch of the first aircraft the coordinates of the launching mechanism can be determined in 7 seconds with an accuracy up to 10−17 m.

An implicit McCormack-type scheme is constructed for the kinetic plasma model based on the VlasovAmpere equations. Compared to the explicit scheme, it has a weaker stability constraint, but retains the same computational efficiency, i.e. it does not use internal iterations. In this case, the error of the total energy corresponds to the second order of accuracy of the algorithm, and the total charge (number of particles) is stored at the grid level. The formation of plasma waves excited by a short powerful laser pulse is considered as a simulated physical process.

Semi-analytical approximations of the tangent derivative (TD) and the normal derivative (ND) of the potential of a simple layer (PSL) near the boundary of a two-dimensional region are proposed, performed within the framework of the collocation method of boundary elements and not requiring approximation of the coordinate functions of the boundary. To obtain approximations, analytical integration over the smooth component of the distance function and a special additive-multiplicative method of distinguishing features are used. It is proved that such approximations have a more uniform convergence near the boundary of the domain compared with similar approximations of TD and PSL ND based on a simple multiplicative method of distinguishing features. One of the reasons for the highly uneven convergence of traditional approximations of the TD and PSL ND based on Gauss’s quadrature formulas has been established.

The article is dedicated to dualism of the soliton solution theories and solutions of functional differential point form equations. The basics of the formalism of such dualism are presented with the central element being the idea of soliton bunch and the dual pair: function-operator. Within such an approach, it is possible to describe the whole space of soliton solutions with a set characteristic as well as their asymptotics both in space and time. An example of traffic flow on the Manhattan grid shows the whole family of limited soliton solutions.

A system of equations with nonlinearity with respect to the potential of the electric field and temperature is proposed, describing the heating process of semiconductor elements of an electric board, and over time, thermal and electrical “breakdowns” may occur. The paper considers a method for the numerical diagnosis of solution destruction. In the process of numerical investigation of this problem, an approach was used based on the reduction of the initial system of partial differential equations to a differential algebraic system, followed by the solution of this system using a one-stage Rosenbrock scheme with a complex coefficient. Numerical diagnostics of the destruction of the exact solution of the specified system of equations was based on the method for calculating a posteriori asymptotically accurate error estimate obtained when calculating an approximate solution on successively thickening grids. Numerical estimates of the moment of destruction are obtained for various initial conditions.

For the transport equation in three-dimensional