Modern methods of direct numerical simulations of transfer processes in random media

The paper considers the methods of direct numerical investigation of the behaviour of dynamical systems of explosive type under the influence of random noise. Dynamical systems are described by a system of nonstationary ordinary differential equations (ODE). The dynamics of the system, taking into account random noise, is described by a system of stochastic ordinary differential equations (SODE). The paper provides an overview of modern algorithms based on modifications of Runge – Kutta integration methods. The features of the analysis of weak and strong convergence of the SODE integration methods are described. Methods for generating random noise with complex temporal structure (color noise) are described. The method of numerical solution of the system of SODE is used to generate random color noise. Examples of the study of the influence of random noise on biological and mechanical systems of explosive type are presented. It is shown that random noises acting on such systems qualitatively change the character of their behaviour.

Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation

In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

Particle Velocity Fluctuations in Viscous Gas with Random Velocity as the Sum of Two Correlated Color Noises

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research.

Stability of Nonlinear Stochastic Distributed Parameter Systems and Its Applications

This paper derives several well-posedness (existence and uniqueness) and stability results for nonlinear stochastic distributed parameter systems (SDPSs) governed by nonlinear partial differential equations (PDEs) subject to both state-dependent and additive stochastic disturbances. These systems do not need to satisfy global Lipschitz and linear growth conditions. First, the nonlinear SDPSs are transformed to stochastic evolution systems (SESs), which are governed by stochastic ordinary differential equations (SODEs) in appropriate Hilbert spaces, using functional analysis. Second, Lyapunov sufficient conditions are derived to ensure well-posedness and almost sure (a.s.) asymptotic and practical stability of strong solutions. Third, the above results are applied to study well-posedness and stability of the solutions of two exemplary SDPSs.