Volume 32, Issue 1

Evolution equations of small disturbances aimed to compute the location of a laminar–turbulent transition in boundary layers by the e N -method taking into account the compressibility and heat transfer are described and justified for aerodynamic applications. The results of computation of N -factors and transition locations with the use of locally-parallel flow assumption for a flat plate exposed to supersonic flow at zero angle of attack are given. The results agree well with known numerical and experimental data.

A two-stage analysis do detect the appearance of a gradient catastrophe of the solution is proposed for quasilinear systems of hyperbolic equations of special form. Applications of the first stage are considered for the following simple cases: scalar Burgers’ equation and a quasi-orthogonal system generalizing it. The entire two-stage analysis is applied to systems of equations describing one-dimensional electron oscillations in plasma, namely, plane oscillations in the relativistic and non-relativistic cases and also axially symmetric non-relativistic cylindrical oscillations.

This paper elaborates a new approach to nonlinear filtering grounded in an accurate implementation of the continuous–discrete extended Kalman filter for estimating stochastic dynamic systems. It implies that the moment differential equations for calculation of the predicted state mean and error covariance of propagated Gaussian density are solved accurately, i.e., with negligible errors. The latter allows the total error of the extended Kalman filter to be reduced significantly and results in a new accurate continuous–discrete extended Kalman filtering method. In addition, this filter exploits the scaled local and global error controls to avoid any comparison of different physical units. The designed state estimator is compared numerically with continuous–discrete unscented and cubature Kalman filters to expose its practical efficiency. The problem of long waiting times (i.e., infrequent measurements) arisen in chemical and other engineering is also addressed.

We propose and investigate numerically two new preconditioners for the matrices, which arise in the mixed-hybrid finite element methods for diffusion equation in strongly heterogeneous media. Both preconditioners include special projectors on the vector spaces orthogonal to the vectors with constant components. We give general description of the preconditioners and discuss numerical results which demonstrate their efficiency compared to the classical diagonal preconditioner.

The influence of numerical approximations on statistical characteristics of modelled two-dimensional turbulence sustained by a stochastic external forcing is studied. The ability of various finite-difference and semi-Lagrangian schemes to reproduce reliably the dual energy and enstrophy cascades for coarse spatial resolution is tested. It is also studied how the requirement of preserving invariants inherent to a two-dimensional ideal fluid is important relative to numerical schemes. The results of calculations with high spatial resolution were taken as a reference solution. The choice of studied schemes was motivated by their use in atmosphere and ocean numerical models, in particular, in the Institute of Numerical Mathematics climate model (INMCM) and semi-Lagrangian absolute vorticity (SLAV) model of medium-range weather forecast. The importance of conservation laws for integral vorticity and enstrophy is revealed in the numerical experiments with a small-scale external forcing.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the observation data is studied. The gradient of the response function is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. The solvability of the non-standard problem is studied, based on the Hessian of the original cost function. An algorithm to compute the gradient of the response function with respect to observations is developed and justified.