Vol 58, No 9 (2018)

The problem of equilibrium distribution of flows in a transportation network in which a part of edges are characterized by cost functions and the other edges are characterized by their capacity and constant cost for passing through them if there is no congestion is studied. Such models (called mixed models) arise, e.g., in the description of railway cargo transportation. A special case of the mixed model is the family of equilibrium distribution of flows over routes—BMW (Beckmann) model and stable dynamics model. The search for equilibrium in the mixed model is reduced to solving a convex optimization problem. In this paper, the dual problem is constructed that is solved using the mirror descent (dual averaging) algorithm. Two different methods for recovering the solution of the original (primal) problem are described. It is shown that the proposed approaches admit efficient parallelization. The results on the convergence rate of the proposed numerical methods are in agreement with the known lower oracle bounds for this class of problems (up to multiplicative constants).

A technique for the numerical simulation of the interaction between an inviscid compressible medium and solid bodies is described. The boundary condition on the solid surface is set using the immersed boundary approach. An immersed boundary technique is proposed for the considered class of problems. The performance of the technique is demonstrated by solving test problems in acoustic scattering.

A hybrid scheme for computing convective fluxes in aeroacoustics simulation is constructed. The scheme combines the upwind and central-difference schemes with a blending factor chosen automatically depending on the flow regime. The performance of the scheme is demonstrated by computing sub- and supersonic airfoil flows.

The balance-characteristic method is used to approximate a two-fluid model for two-phase gas–liquid flows. The original system of equations is a hyperbolic one with different phase pressures and an additional equation for pressure relaxation. A finite-difference approximation is constructed for the system. Numerical results for the motion of a gas–liquid mixture are obtained and compared with results of standard numerical schemes.

Many processes in complex systems are nonlocal and possess long-term memory. Such problems are encountered in the theory of wave propagation in relaxing media [1, p. 86], whose equation of state is distinguished by a noninstantaneous dependence of the pressure p(t) on the density ρ(t); the value of p at a time t is determined by the value of the density ρ at all preceding times; i.e., the medium has memory. Similar problems are also encountered in mechanics of polymers and in the theory of moisture transfer in soil [2]; the same equation arises in the theory of solitary waves [3] and is also called the linearized alternative Korteweg–de Vries equation, or the linearized Benjamin–Bona–Mahony equation. One of such problems was studied in [4]. In the present paper, a locally one-dimensional scheme for parabolic equations with a nonlocal source, where the solution depends on the time t at all preceding times, is considered.

Sufficient conditions of the onset of the blow-up mode for the parabolic equation with double nonlinearity that describes the combustion process in a nonlinear medium are analyzed. A sufficient condition for the initial function under which the time of the solution blow-up of the corresponding initial–boundary value problem for this equation is finite is obtained. An analytical upper bound on the solution existence time is derived. This a priori bound on the solution blow-up time is used for the numerical refinement of the blow-up process. It is shown that the numerical diagnostics of the solution blow-up makes it possible to obtained a better estimate of the solution blow-up time and detect the fact of local blow-up with respect to the spatial variable.

A mixed problem for a homogeneous wave equation with fixed ends, a summable potential, and a nonzero initial velocity is studied. Using the resolvent approach and developing the Krylov method for accelerating the convergence of Fourier series, a classical solution is obtained by the Fourier method under minimal conditions on the smoothness of the initial data and a generalized solution in the case of the initial velocity represented by an arbitrary summable function is found.