Volume 35, Issue 2

The paper is focused on problem of filtering random processes in dynamical systems whose mathematical models are described by stochastic differential equations with a Poisson component. The solution of a filtering problem supposes simulation of trajectories of solutions to a stochastic differential equation. The trajectory modelling procedure includes simulation of a Poisson flow permitting application of the maximum cross section method and its modification.

Eddy-permitting numerical ocean models resolve mesoscale turbulence only partly, that leads to underestimation of eddy kinetic energy (EKE). Mesoscale dynamics can be amplified by using kinetic energy backscatter (KEB) parameterizations returning energy from the unresolved scales. We consider two types of KEB: stochastic and negative viscosity ones. The tuning of their amplitudes is based on a local budget of kinetic energy, thus, they are ‘energetically-consistent’ KEBs. In this work, the KEB parameterizations are applied to the NEMO ocean model in Double-Gyre configuration with an eddy-permitting resolution (1/4 degree). To evaluate the results, we compare this model with an eddy-resolving one (1/9 degree). We show that the meridional overturning circulation (MOC), meridional heat flux, and sea surface temperature (SST) can be significantly improved with the KEBs. In addition, a better match has been found between the time power spectra of the eddy-permitting and the eddy-resolving model solutions.

The analysis of the modified partial differential equation (MDE) of the constant wind speed advection equation explicit difference scheme up to the eighth order with respect to both space and time derivatives is presented. So far, in majority of publications this modified equation has been derived mainly as a fourth-order equation. The MDE is presented in the so-called Π -form of the first differential approximation. This form includes only the space derivatives of higher order p and their coefficients μ ( p ). Analysis of these coefficients provides indications of the nature of the dissipative and dispersive errors. A fragment of the stencil for determining the modified differential equation up to the eighth-order MDE for the second-order Beam–Warming scheme is included. The derived coefficients μ ( p ) as well as the analysis of the phase shift errors, the phase and group velocities and dispersive features on the basis of these coefficients have not been published so far. The dissipative features of this method we present in [33].

The problem of guaranteed computation of all steady states of the Marchuk–Petrov model with fixed values of parameters and analysis of their stability are considered. It is shown that the system of ten nonlinear equations, nonnegative solutions of which are steady states, can be reduced to a system of two equations. The region of possible nonnegative solutions is analytically localized. An effective technology for computing all nonnegative solutions and analyzing their stability is proposed. The obtained results provide a computational basis for the study of chronic forms of viral infections using the Marchuk–Petrov model.

A mathematical model of the dynamics of cardiomyocyte death in myocardial infarction during the acute phase of the disease is developed. An economical computing technology of structural and parametric identification of model equations is presented based on the use of experimental data as dynamic parameters and on the idea of splitting the inverse coefficient problem with a large number of unknown parameters into a sequence of simpler inverse problems. The relevance of the proposed mathematical model is confirmed by comparison of the numerical solution to the problem with experimental data and also with the results of modelling of known therapeutic strategies.