Computational Mathematics and Mathematical Physics

We consider singularities of the wave front of a smooth plane curve from the viewpoint of planar vector fields and their singularities. This approach allows to give complete description of wave fronts, caustics, and their singularities in generic cases as well as in degenerate ones. We give explicit formulas for asymptotics in the neighborhood of singular points of wave fronts and caustics. We study some bifurcations of wave fronts and caustics under deformations of the curve.

The paper discusses stationary and parabolic turning point problems with a variable diffusion coefficient and a small parameter. It constructs layer-eliminating coordinate transformations and corresponding layer-resolving grids, and analyzes convergence of numerical solutions through the upwind and second-order schemes on the layer-resolving grids.

It is shown that a strongly convex Lipschitz differentiable function satisfies the Polyak-Lojasiewicz condition on a proximally smooth C¹-smooth manifold for a certain relationship of the constants of proximal smoothness of the manifold and strong convexity of the function. The mentioned condition guarantees a linear rate of convergence of the gradient projection method for minimizing a function on a manifold. An algorithm is proposed for finding the metric projection of a point located sufficiently close to a manifold onto this manifold.

The global solvability and local uniqueness of the solution to an inhomogeneous boundary value problem for a model of electron drift–diffusion in polar dielectrics are proven. A maximum–minimum principle is established for the charge density. The results of a finite-element implementation of a mathematical model for the charging process of polar dielectrics under electron irradiation are presented and discussed.

The paper compares the results of solving radiation transport problems in a medium using the diffusion approximation (P ₁ approximation) and without using it. Bicompact schemes constructed by the method of lines on a minimal two-point stencil are used for numerical solving of the problems. Schemes have the fourth order of approximation in space. Three-stage L-stable Runge-Kutta method of the third order of approximation is used for time integration to solve radiation transport problems in a homogeneous medium. The implicit Euler method of the first order of approximation is used for time integration to solve radiation transport problems in a heterogeneous medium. The quasi-diffusion method of V. Ya. Gol'din which is one of the variants of HOLO algorithms is used to accelerate the convergence of iterative processes. A joint solution of high order (HO) and low order (LO) equations (transport equation and system of quasi-diffusion equations respectively) is organized. The dimensionality of the problem is reduced twice by averaging over the angular and energy variables. The schemes for the low order equations are kinetically consistent with the scheme for the high order transport equation. The schemes are implemented and applied to solve the first and second Fleck problems in a multigroup approximation. An algorithm for monotonization of schemes is proposed and tested. The hybridization of solutions obtained using schemes of high and low orders of approximation is performed on each stage of the HOLO algorithm. A comparison of calculations using the full HOLO algorithm, which includes the solution of the transport equation, and calculations in the diffusion approximation was conducted. Fleck problems are characterized by the absence of a line spectrum, in that case the difference in the group Planck and Rosseland absorption coefficients usually do not exceed one order of magnitude. The difference can reach several orders of magnitude for problems with a line spectrum. It is shown that the differences in the solutions of Fleck problems can be noticeable at small calculation time. The differences in the solutions decrease as the calculation time increases. The stationary solutions obtained by the two methods are practically the same for the solved problems. The greatest differences occur at short times in the optically dense region. If a small loss of calculation accuracy is acceptable and there is no need to represent all the details of the processes development at small times, then the use of the diffusion approximation is justified.

The mathematical model of a two-dimensional electrostatic problem for the cross section of an electric cable is considered. In the model, the seeking quantities are distributed electrical potential outside conductors and constant potential at the boundary of each conductor. Known quantities are the distributed electrical charge in isolation and the full charge of each conductor. To solve a boundary value problem with unknown potential conditions on the conductor boundaries, the article studies a method that replaces the specified problem by a set of conventional Dirichlet boundary problem. The unique resolution of the boundary electrostatic problem is shown and a detailed algorithm for its solution is described. The Galerkin finite element algorithm with quadratic elements is adapted for numerical modeling of a spatially charged electrical cable. The method is suitable for calculating multicore cables with any number of wires and arbitrary geometry. The results are presented, demonstrating the workability of a numerical method for solving electrostatic problems with potential conditions on conductors. The technique can be used to model electrical disturbances in spatially charged multicore cables.

Hydrodynamic instabilities pose a serious challenge when optimizing devices designed to achieve high energy densities. This paper uses two algebraic models and the ILES methodology to investigate the development of the Richtmyer-Meshkov instability from its origination to the transition to mixing the contacting gases. A two-layer gas system with a sinusoidally disturbed contact boundary is considered. The simulation is performed for three Atwood numbers (A = 0.2, 0.54, 0.82) at three Mach numbers (M = 1.3, 3, 5) of the shock wave incident on the contact boundary.