Volume 26, Issue 4

For a given function, we consider the problem of minimizing the P 1 interpolation error on a set of triangulations with a fixed number of triangles. The minimization problem is reformulated as the problem of generating a mesh which is quasi-uniform in a specially designed metric. For functions with indefinite Hessian, we show the existence of a set of metrics with highly diverse properties. This set may include both anisotropic and isotropic metrics, which produce families of different meshes providing a comparable reduction of interpolation error. The developed theory is verified with numerical examples.

Solutions to initial boundary value problems are constructed for the shallow water equations in the form of series locally convergent in the neighbourhood of a movable water–land boundary for an arbitrary bottom relief. The motion law and the velocity of this boundary are determined for various wave–shore interaction modes. The obtained results of analytic study of the solutions are used for the development of approximations of boundary conditions on the movable shoreline. Test problems are numerically solved using an explicit predictor-corrector scheme of the second order of approximation on adaptive grids retracing the position of the water–land boundary. The results of these calculations are presented.

The effect of non-axial breaking of relativistic and nonrelativistic cylindrical axially-symmetric nonlinear plasma oscillations was earlier identified and then numerically and analytically studied by the authors within a one-dimensional hydrodynamic model. In this paper, an explicit method of the second order of accuracy (on smooth solutions) is proposed and implemented in Euler variables for the two-dimensional simulation of the indicated effect. The results of calculations admitting comparison with the one-dimensional model are presented.

We prove a theorem on the existence of periodic trajectories in phase portraits of odd-dimensional nonlinear dynamical systems modelling gene networks regulated by negative feedbacks. Also, we find certain sufficient conditions of the existence of stable cycles there. Some generalizations and applications of these results are given as well.

We study the effects caused by inexact knowledge of the Poisson's ratio in the linear elasticity problem and derive estimates for the incremental quantity that characterizes the variability of the internal energy and the logarithmic derivative of the energy with respect to the Poisson's ratio. These quantities characterize errors caused by inexact (incomplete) knowledge of material constants, which is typical for computer modelling of physical and engineering problems.We prove that their behaviour is drastically different for two different classes of boundary conditions, namely the boundary conditions of the first class generate solutions that are relatively stable with respect to small variations of the Poisson's ratio, while other conditions generate solutions, for which the quantities blow up if the ratio tends to one half.

Grid approximations of the Dirichlet problem are considered in a vertical strip for the semilinear elliptic convection–diffusion equation; for this problem, the nonlinear difference scheme based on the classic approximation of the problem on a piecewise-uniform grid refined in a layer converges ε -uniformly in the uniform norm with the convergence rate order not higher than one. Using the Richardson technique, we construct a nonlinear scheme ( the Richardson scheme on nested piecewise-uniform grids ) convergent ε -uniformly with the improved convergence rate , where N 1 +1 and N 2 +1 are the numbers of nodes on the axis x 1 and on the unit segment of the axis x 2 , respectively. In order to solve this scheme, we construct iterative schemes of higher accuracy , which are the linearized scheme (the nonlinear term is calculated based on the desired function taken from the previous iteration) and the scheme of the Newton method , as well as truncated variants of iterative schemes convergent at the rate . The number T of iterations required in the truncated schemes for the solution of the problem does not depend on the parameter ε , here T = 𝒪(ln (min[ N 1 , N 2 ])) and T = 𝒪(ln ln (min[ N 1 , N 2 ])) in the case of the truncated linearized scheme and the truncated Newton scheme, respectively.