Volume 36, Issue 3

New solution algorithms of optimal filtering problem are proposed for systems with random structure and continuous time. This problem consists in estimating the current state of system based on the results of measurements. The mathematical model of the system includes nonlinear stochastic differential equations whose right-hand side determines the structure of the dynamic system or mode of operation. The right-hand side may vary at random time moments. The number of structures of the system is assumed to be finite and the process of changing the structure to be Markov or conditionally Markov. The state vector of such system consists of two components, namely, a vector with real coordinates and an integer structure number. The law of change of the structure number is determined by the distribution of the random time interval between switchings with a given intensity dependent on the state of system.

High precision simulation algorithms are proposed and justified for modelling cold plasma oscillations taking into account electron–ion collisions in the non-relativistic case. The specific feature of the approach is the use of Lagrangian variables for approximate solution of the problem formulated initially in Eulerian variables. High accuracy is achieved both through the use of analytical solutions on trajectories of particles and due to sufficient smoothness of the solution in numerical integration of Cauchy problems. Numerical experiments clearly illustrate the obtained theoretical results. As a practical application, a simulation of the well-known breaking effect of multi-period relativistic oscillations is carried out. It is shown that with an increase in the collision coefficient one can observe that the breaking process slows down until it is completely eliminated.

A new mathematical model of the diffusion–convective process with ‘memory along the flow path’ is proposed. This process is described by a homogeneous one-dimensional Dirichlet initial-boundary value problem with a fractional derivative along the characteristic curve of the convection operator. A finite-difference approximation of the problem is constructed and investigated. The stability estimates for finite-difference schemes are proved. The accuracy estimates are given for the case of sufficiently smooth input data and the solution.

This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.

In this paper we consider the problem of modelling a system of aggregating particles, that are being transported with stationary velocities dependent on masses of the particles in one-dimensional case. A numerical method based on the ideas of POD (Proper Orthogonal Decomposition) is constructed, and its capacity to speed up the solution up to 40 times is demonstrated.