Vol 52, No 7 (2016)

We carry out an analysis of hyperbolized equations of diffusion type convenient for modeling on high-performance computer systems; in particular, we study conservation laws for these equations.

We study the effect of the approximation viscosity of Godunov-type difference schemes in a model problem on the accretion disk in a binary star system. Computations are carried out by schemes of the first and increased approximation order. In the higher-order scheme, we take into account the viscous stress tensor, which is used to model the influence of the approximation viscosity observed in the first-order scheme. It is noteworthy that the approximation viscosity can lead to global qualitative changes in the flows to be modeled. The numerical results given in the paper are of independent interest in that they illustrate some new specific features of gasdynamic flows in this classical problem of modern astrophysics.

We consider a mathematical model of equilibrium configurations of plasma, magnetic field, and electric field in a toroidal trap with two ring conductors with current loaded into plasma. We present the mathematical apparatus of the model based on the numerical solution of boundary value problems for the Grad–Shafranov equation (a differential equation of elliptic type for the magnetic flux function), solution methods for these problems, and numerically obtained properties of equilibrium configurations. We indicate the differences in configurations in the toroidal trap and in its analog straightened into a cylinder.

We study a linear integro-differential equation with a coefficient that has zeros of finite order. For its approximate solution in the space of generalized functions, we suggest and justify special generalized cases of the method of subdomains, the collocation method, and the moment method.

We construct a mathematical model of electromagnetic processes in a magnetic accelerator. In the two-dimensional approximation, the Maxwell equations are reduced to a system of scalar integro-differential equations in the conductors and to the Laplace equation in the dielectric subdomains. We obtain a numerical model on the basis of the Galerkin–Petrovmethod with piecewise constant and piecewise linear basis functions. The results of computations are represented.

For stationary linear convection–diffusion problems, we construct and study a new hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability, accuracy, and superconvergence of the schemes. A new procedure for the post-processing of solutions of HDG-schemes is suggested.

We study unbounded solutions of a broad class of initial–boundary value problems for multidimensional quasilinear parabolic equations with a nonlinear source. By using a conservation law, we obtain conditions imposed solely on the input data and ensuring that a solution of the problem blows up in finite time. The blow-up time of the solution is estimated from above. By approximating the source function with the use of Steklov averaging with weight function coordinated with the nonlinear coefficients of the elliptic operator, we construct finite-difference schemes satisfying a grid counterpart of the integral conservation law.