Том 59, № 10 (2019)

To preserve some invariant properties of the original differential equation is an important criterion to judge the success of a numerical simulation. In this paper, we construct, analyze and numerically validate a class of momentum-preserving finite difference methods for the Korteweg-de Vries equation. The proposed schemes can conserve the discrete momentum to machine precision. Numerical experiments reveal that the phase and amplitude errors, after long time simulation, are well controlled due to the momentum-preserving property. Besides, the numerical results show that the numerical errors grow only linearly as a function of time.

An efficient method combining classical (gradient-based) methods for global scalar optimization and genetic algorithms for multiobjective optimization (MOO) is proposed for approximating the Pareto frontier and the Edgeworth–Pareto hull (EPH) of the feasible objective set in complicated nonlinear MOO problems involving piecewise constant functions of criteria with numerous local extrema. An optima injection method is proposed in which the global optima of individual criteria are added to the population of a genetic algorithm. It is experimentally shown that the method is significantly superior to the original genetic algorithm in the order of convergence and the approximation accuracy. Experiments concerning EPH approximation are also performed for the problem of constructing control rules for a cascade of reservoirs with criteria reflecting the reliability with which the requirements imposed on the cascade are met.

Open-loop control of a plant governed by a system of differential equations in normal form is studied. The right-hand side of this system includes nonsmooth terms. The original problem is reduced to the unconstrained minimization of a nonsmooth functional. Necessary and sufficient minimum conditions in terms of subdifferential are found. Based on these conditions, the subdifferential descent method is used to solve the problem. The proposed approach is illustrated by numerical examples.

Statistical properties of closed biocenoses described by a Volterra chain subject to periodic boundary conditions are studied. The periodicity of the boundary conditions makes the interaction matrix singular. This causes certain difficulties in statistical description. The system dynamics is described by a class of Hamiltonians. The change of Hamiltonians within the class preserves the system dynamics, but it changes its statistical properties. Hence, the problem of selecting a unique Hamiltonian that correctly describes the statistical properties of the system. This problem is solved, and the statistical properties of the system are described.

The stability of the flow of an incompressible polymer fluid with a strong discontinuity when the fluid can flow through the discontinuity is discussed. Particular solutions of the linearized problem that increase with time are constructed numerically.

The formulation and applicability range of the \(\gamma \)-model of laminar–turbulent transition are analyzed. Its implementation in the EWT-TsAGI application package and verification results for the problem of computing a transitional boundary layer on a plate are described. The cases of zero and adverse pressure gradients are considered. A modification of the \(\gamma \)-model that takes into account the effect of compressibility on the laminar–turbulent transition location is proposed. Two problems are considered to calibrate and test the correction term: the flow over the LV6 laminar-flow airfoil and the flow over the nacelle of a bypass turbojet engine. It is shown that, with the used of the proposed correction, the laminar–turbulent transition location can be correctly predicted in transonic flow regimes.

It is well known that the steady-state plane-parallel or spatial axisymmetric flow of an ideal incompressible fluid in a finite-length plane channel or pipe that can be decomposed in powers of spatial coordinates (i.e., is an analytical and, hence, exactly computable flow) is uniquely determined by the inflow vorticity. Under the same boundary conditions, an infinite number of uncomputable phantoms, i.e., infinitely smooth, but nonanalytical flows exist if the domain of a unique analytical flow contains a sufficiently intense vortex cell where the maximum principle is violated for the stream function. A scheme for obtaining an uncomputable vortex phantom for the Euler fluid dynamics equations is described in detail below.