Volume 28, Issue 4

In this paper we develop efficient numerical schemes to solve ordinary differential equations. Our methods are of the Nordsieck type, adaptive and capable of automatically controlling the global error of a numerical solution. A special feature of the new stepsize selection algorithms introduced here is the global error estimation quality control. Two different ways of attaining the preassigned accuracy of computation are examined in the paper. Namely, we implement the global error control mechanism based on reducing the maximum stepsize bound and the other one is based on reducing the local error tolerance. An accurate starting procedure for the adaptive Nordsieck methods is presented in full detail. Our intention here is to find the most effective strategy of stepsize selection. Theoretical investigation is supplied with numerical tests.

The paper presents a new global three-dimensional model of the Earth thermosphere (for altitudes from 90 to 500 km) with a high spatial resolution. The model uses simple approximations for calculation of solar radiation and ion-neutral interaction. A detailed description of the algorithm is presented. It is shown that the model represents the main features of the general thermospheric circulation with a satisfactory accuracy.

An approach to the study of the conditioning of difference schemes and their stability to data perturbations is developed for the Dirichlet problem for a singularly perturbed convection-diffusion ordinary differential equation with the perturbation parameter e, e 2 (0;1]. We consider a standard difference scheme, which is a monotone scheme on a uniform grid, and a special scheme of the grid solution decomposition method. Constructing the special scheme, we use a decomposition of the grid solution into a regular and a singular components that are the solutions to grid subproblems considered on uniform grids. The following facts are proved for the standard scheme: (a) the scheme converges only for N -1 = o(ε) at the rate (N -1 (ε +N -1 ) -1 ), , where N +1 is the number of grid nodes, (b) the scheme is not e-uniformly well conditioned and stable to perturbations, (c) the condition number of the scheme has the order (ε -1 δ -2 , where δ is the accuracy of the grid solution, δ = δst ≈ N -1 (ε +N -1 ) -1 . In the case of convergence of the standard scheme proved theoretically, the actual accuracy of the computed solution decreases with the decrease in the parameter e under the presence of perturbations and may completely vanish for a sufficiently small e (namely, under the condition t = (lnε -1 +lnδ -2 st ), where t is the number of digits in the machine word). At the same time, the special scheme of the solution decomposition method converges e-uniformly in the uniform norm at the rate (N -1 lnN); in the variables e and d, the special scheme is e-uniformly well conditioned and stable to grid problem data perturbations; the condition number of the scheme of the decomposition method is of the order (δ -2 lnδ -1 ), δ = δdec ≈ N -1 lnN.

Recently, ensemble Kalman filters have come into practical data assimilation for numerical weather prediction models. We give an overview of ensemble Kalman filters and problems that arise with practical implementation of ensemble methods. We present our implementation of the local ensemble transform Kalman filter, one of ensemble square root filters using observation localization. Multiplicative and additive inflations are used to prevent filter divergence and to account for the model error. The implemented assimilation system is tested with the global semi-Lagrangian atmospheric model SL-AV using real observations for 2 months of cyclic assimilation (August and September 2012). The system works stably. Application of the ensemble filter significantly reduces first guess (background) errors and corrects the forecast biases.