Volume 19, Issue 3

In this study, we apply optimal control approaches and adjoint equations to construct iterative processes for some problems of hydrodynamics. First, the iterative processes are proposed for an abstract problem given by the system of operator equations. Second, the general methodology is applied to the generalized Stokes equations perturbed by a skew-symmetric bounded operator and to the stationary linearized Navier–Stokes equations (the Oseen problem). The convergence rates of the iterative algorithms are studied.

We develop new weight modifications of the method of direct statistical simulation for the approximate solution of the nonlinear Boltzmann equation by using the ingenious procedure proposed by the authors, viz. the stratification of collision distribution in a many-particle system by the number of a pair of interacting particles. We check the appropriate weight algorithms for the model problem with the known solution. They allow us to estimate the variations of functionals under small changes of parameters.

We consider the problem of optimal control by a boundary condition for the heat equation. This problem arises in the process of mathematical formulation of a data assimilation problem for the reconstruction of the ocean surface temperature by approximate initial data and a solution. For the regularized functional, existence and uniqueness of a solution to this problem is proved; an efficient iterative method for solving this problem is proposed and justified.

This paper deals with the construction of stable infinitely extendable Chebyshev iterative processes and explicit stable methods for solving stiff systems of differential equations. It is a continuation of [V. I. Lebedev and S. A. Finogenov, On construction of the stable permutations of parameters for the Chebyshev iterative methods. Part 1. Russ. J. Numer. Anal. Math. Modelling (2002) 17 , No. 5, 437–456.]. In the present paper the authors prove for the general case the estimates of values affecting stability, which are independent of the length of the iterative process.

We define the regularly structured pseudospectrum of a square complex matrix and describe its main properties. Using a difference one-dimensional convection-diffusion operator as an example, we explain why the regularly structured pseudospectrum is better than the ordinary pseudospectrum when we are interested in the question of how the minimal-in-magnitude eigenvalues of the finite-dimensional approximation of a differential operator can vary when its coefficients are perturbed.

We present and justify difference schemes for obtaining the solution of the Dirichlet problem, its first derivatives and pure second derivatives on a cubic grid with uniform accuracy O ( h 2 ), h is a grid step. We propose a method for finding a mixed derivative with accuracy O ( h 2 / ( ρ + h )), where ρ is the distance from a current mesh node to the boundary of a parallelepiped.