Vol 8, No 4 (2016)

Boundary layers often appear at the interface of two media. A typical example is the singularly perturbed Helmholz equation. It is shown that the modern finite difference methods solve such problems with a high efficiency under an appropriate choice of the grid. A convergence verification procedure is proposed that does not require majorant estimation. A superfast algorithm yielding an a posteriori asymptotically precise error estimate is described and a quasi-uniform rectangular grid resolving in detail all the segments of the solution is proposed. The algorithm guarantees good precision even on the moderate grids having N ~ 200 points in each direction. This algorithm is implemented in MATLAB.

The features and basic functionality of the MVTU software package are considered. The software is intended for the research and design of a wide range of systems described by differential, algebraic, and difference equations. Examples of the solutions of the research and applied problems are given.

The present paper puts forward a mathematical model of laser radiation absorption in a laser target, which combines approximations of geometrical and wave optics and the corresponding numerical algorithm. This model depends on the principles of geometrical optics in the range of weak variation of the plasma refractive index on the scale of the wave length. This enables one to describe the refraction of the radiation. A transition to wave approximation is carried out near the surface of critical density, where the approximation of the geometrical optics is a fortiori inapplicable. For this aim, the plasma medium is approximately represented as a set of plane layers, on which the one-dimensional Helmholtz equation is solved. This makes it possible to construct a simple and relatively accurate method for calculating the absorption and reflection of laser radiation near the critical density surface in order to effectively take into account the dependence of the interaction of radiation with matter on the polarization direction, etc. The proposed model is adapted for implementation in the radiation gas dynamics (RGD) code. A numerical computation subroutine is presented based on the analytical solution of the differential equations corresponding to the optical ray model of the laser radiation energy flux. This solution is obtained under the assumption that the squared gradient of the refractive index is constant in any cell of the mesh. The convergence rates of the proposed algorithms are estimated using the data obtained in the numerical experiments.

In this paper, a mathematical model is developed for information warfare in society whereby an individual chooses between two suggested viewpoints. The model is based on the traditional Rashevsky framework of imitative behavior. A primary analysis of the model is conducted. The model has the form of a nonlinear integro-differential equation in which the unknown function is under the sign of the derivative and within the integration limit and acts as an argument of an exogenously given function.

We present mathematical models and methods of the computational experiment using a three-dimensional RMHD model simulating the implosion of Z-pinches formed by an imploding cylindrical array of thin tungsten wires. The calculations take into account the discrete structure of the array and the extended plasma formation at the evaporation of the wire material and they yield the estimated values of the output power and X-ray spectrum. The data are presented on the spatio-temporal distribution in the parameters of the Z-pinch plasma, including the velocity, electron and ion temperatures, ionization degree, and the power output integrated over space. We have also calculated the characteristics of the Z- pinch X-ray spectrum depending on the photon energy at different times after the discharge current starts. It has been found that the trailing mass of tungsten on the periphery affects the emission intensity of the central part of the pinch in the radial direction, which can be explained by the emission absorption in the peripheral plasma layers of the trailing tungsten mass. A detailed model has been constructed of the central pinch formed by the electric current implosion of the material of multiwire tungsten arrays. This model enables one to calculate the intensity of the soft X-ray emission with a temporal, spatial, angular, and spectral resolution for specific experiments on the Angara-5-1 experimental complex intended to study the implosion of cylindrical multiwire arrays, for which there is sufficient information about the time profile of the absolute emission intensity in the low-energy range of the X-ray emission. The obtained numerical results can be directly compared with the experimental values. The RMHD model simulating the implosion of Z-pinches has been verified by comparing its results with the experimental implosion indicators.

Being applicable for high-performance computing systems, a parallel code based on the sequential QuDiff code was developed to calculate the critical assembly of the fast neutron reactor active zone. The multigroup transport equation calculation method was based on V. Ya. Goldin’s quasi-diffusion method. For efficient algorithm construction, it was suggested to use all the possible reactor assembly symmetries for the self-adjustable neutron-nuclear operation mode. MPI was applied as a parallel interface. The domain decomposition method was used. The pipelined parallelization method was applied for the consistent parallelization of the calculation of the quasi-diffusion system of equations and for the calculation of the transport equation. The calculations performed for the 3D active zone model of the BN-800 reactor capable of operating in a self-adjustable neutron-nuclear mode showed that the parallel code QuDiff is highly scalable. The present results can be used in the dynamic numerical simulation of a fast reactor’s active zones.

An algorithm for constructing the stiffness matrix of a quadrangular curvilinear finite element in the form of a fragment of the middle surface of an elliptic cylinder with 18 degrees of freedom in the node is given. Implementation of the finite-difference procedure includes two variants of approximation of the sought variables, i.e., a scalar one and a vector one. Numerical examples are given and prove that the vector approximation has principal advantages over the scalar approximation for arbitrary shells with significant gradients of curvature of the lines of the middle surface or allowing for displacement as a rigid body.