Vol 55, No 7 (2019)

The notion of stiffness of a system of ordinary differential equations is refined. The main difficulties encountered when solving the Cauchy problem for stiff systems are indicated. The advantages of switching to a new argument, the integral curve arc length, are demonstrated. Various mesh step selection criteria are discussed, and the integral curve curvature criterion is recommended. The most reliable implicit and explicit schemes suitable for solving stiff problems are presented. A strategy permitting an asymptotically accurate computation of the error of a numerical solution simultaneously with the solution itself is described. An analysis of chemical kinetics of hydrogen combustion in oxygen with allowance for 9 components and 50 reactions between them is provided as an illustration.

We consider a mathematical model of plasma flows in nozzle channels formed by two coaxial electrodes. The acceleration of plasma in an azimuthal magnetic field is accompanied by its compression and heating in the compression zone at the channel axis past the tip of the shorter central electrode. The mathematical apparatus of the model is based on numerically solving two-dimensional MHD problems using the Zalesak flux-corrected transport (Z-FCT) scheme. In the computations, we study the dependence of the compression phenomenon and its quantitative characteristics on the channel geometry, the problem parameters, and the additional longitudinal magnetic field present in the channel.

We propose a generalization of methods for solving problems for a linear elliptic operator with a known fundamental solution in an unbounded domain to the case of mixed-type problems. Two new methods are constructed, the method of setting integral boundary conditions and a three-stage iterative method. The scope of the methods constructed is limited to the cases where the operators have a known fundamental solution outside some finite domain. Numerical algorithms implemented as computer software are created to solve particular problems with the methods proposed. The applicability of the methods to differential and integro-differential equations in the two-dimensional case and the possible applicability to three-dimensional problems are demonstrated.

We consider the Cauchy problem for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problem of finding an unknown function that is a coefficient of the equation and also occurs in the initial condition is posed. The values of the solution of the Cauchy problem and its derivative at x = 0 are given as additional information for solving the inverse problem. An iterative method for determining the unknown function is constructed, and its convergence is proved. Existence theorems are proved for the solution of the inverse problem.

A mathematical model is proposed for shielding permanent magnetic fields by a cylindrical thin-walled multilayered film screen made of materials with permeability depending nonlinearly on the magnetic field intensity. Boundary value problems with Robin boundary conditions and integral boundary conditions on the screen surface are stated for the magnetostatic equations. Efficient numerical methods are developed for solving these problems with allowance for the matching of contact layers with different magnetic properties. A new form of the nonlinear permeability factor is derived from experimental data. The magnetic field potential and intensity in the multilayered screen, as well as the shielding efficiency factor, which characterizes the attenuation of an external magnetic field when penetrating into the cylindrical screen, are studied numerically.

We describe a finite element method for solving 3D problems of nonlinear elasticity theory in the framework of finite strains for a hyperelastic material. Constitutive equations written with the use of the polar and upper triangular (QR) decompositions of the deformation gradient are considered. Our method permits developing an efficient, easy-to-implement technique for the numerical analysis of the stress—strain state of any hyperelastic material.