Band 37, Heft 1

A Monte Carlo algorithm based on the use of transfer matrices is developed to describe the stochastic dynamics of the rotational--translational motion of aerosol clusters taking into account fluctuations in the molecular fluxes of the gas medium. In the general case, the cluster is immersed into a rarefied gas medium, the temperatures of its surfaces may differ from the temperature of the surrounding gas, for example, due to absorption of visible and infrared radiation. The motion of the cluster is described based on Langevin motion equations. The algorithm allows one to calculate parameters of the probability distribution of a six-dimensional vector consisting of components of the momentum and angular momentum vectors transmitted to the cluster by molecular flows. The numerical method allows one to apply preliminary analytical averaging modulo velocities of molecules for both the average values of the components of momentum and angular momentum and their correlation characteristics, which significantly reduces the calculation time.

This work is devoted to the constant (time-independent) upper bounds on the function ∥ exp( tA )∥ 2 where t ⩾ 0 and A is a square matrix whose eigenvalues have negative real parts. Along with some constant upper bounds obtained from known time-dependent exponential upper bounds based on the solutions of Lyapunov equations, a new constant upper bound is proposed that has significant advantages. A detailed comparison of all these constant upper bounds is carried out using 2 × 2 matrices and matrices of medium size from the well-known NEP collection.

New methods for constructing an approximation of the diffusion operator for the two-dimensional equation of the ambipolar diffusion process in the F layer of the Earth's ionosphere are presented. This equation is solved in the framework of modelling the global thermosphere and ionosphere dynamics (for the altitudes from 90 to 500 km). The proposed schemes have finite-difference versions of the integral identity, which is a property of differential diffusion equation and which represents the geometric properties of the process (diffusion proceeds along the magnetic field lines of the Earth). The stability of the proposed schemes is analyzed, as well as the accuracy estimates are obtained on the base of the model analytical solution and during the calculations with physically realistic data. A comparison is made with the second-order finite-difference scheme developed earlier for solving the same problem.

Prediction of pollution in water flow is a very important task. To this end, it is imperative to be able to define the uncertainty in a model prediction. This is the purpose of sensitivity analysis whose role is to identify what uncertainty in the model outputs is attributable to the model inputs (parameters in this case). Traditionally, this is achieved by running the model perturbed by many random samples in the parameter space to determine their impact on the model outputs. It provides information on how much of the output variance is controlled by each parameter of the inputs. The theoretical results related to the procedure based adjoint approach for computing a sensitivity of the response function (RF) to changes in the input source are presented in the paper. It is shown that this approach allows to compute, by one single integration of the adjoint equation over a given time interval, a sensitivity of the RF to any source located in the domain of interest. The proposed approach is applied to the 2D Saint-Venant flow equations for modelling the water pollution problem. A numerical experiment is formulated and implemented for the Thanh Nhan Lake in Hanoi for studying a sensitivity of some RF to observations. The numerical model is constructed by applying the well-known finite-volume method. Two appropriate optimization problems are introduced and solved on the basis of the BFGS algorithm. The numerical results show the efficiency of the proposed method and confirm the theoretical findings.

In the present paper we utilize the Proper Orthogonal Decomposition (POD) method for model order reduction in application to Smoluchowski aggregation equations with source and sink terms. In particular, we show in practice that there exists a low-dimensional space allowing to approximate the solutions of aggregation equations. We also demonstrate that it is possible to model the aggregation process with the complexity depending only on dimension of such a space but not on the original problem size. In addition, we propose a method for reconstruction of the necessary space without solving of the full evolutionary problem, which can lead to significant acceleration of computations, examples of which are also presented.