Vol 58, No 6 (2018)

Using properties of the difference schemes approximating a one-dimensional transport equation as an example, it is shown that optimization of the properties of difference schemes based on the analysis in the space of undetermined coefficients and optimization of these properties based on the method of parametric correction are dual problems. Hybrid difference schemes for the linear transport equation are built as solutions to dual linear programming problems. It is shown that Godunov’s theorem follows from the linear program optimality criterion as one of the complementary slackness conditions. A family of hybrid difference schemes is considered. It is shown that Fedorenko’s hybrid difference scheme is obtained by solving the dual linear programming problem.

The problem under study is, given a finite set of vectors in a normed vector space, find a subset which maximizes the norm of the vector sum. For each \({{\ell }_{p}}\) norm, \(p \in [1,\infty )\), the problem is proved to have an inapproximability bound in the class of polynomial-time algorithms. For an arbitrary normed space of dimension \(O(logn)\), a randomized polynomial-time approximation scheme is proposed.

The properties of a family of new adaptive symplectic conservative numerical methods for solving the Kepler problem are examined. It is shown that the methods preserve all first integrals of the problem and the orbit of motion to high accuracy in real arithmetic. The time dependences of the phase variables have the second, fourth, or sixth order of accuracy. The order depends on the chosen values of the free parameters of the family. The step size in the methods is calculated automatically depending on the properties of the solution. The methods are effective as applied to the computation of elongated orbits with an eccentricity close to unity.

Problems that can be reduced to polynomial and parametrized linear matrix inequalities are considered. Such problems arise, for example, in control theory. Well-known methods for their solution based on a search for nonnegative polynomials scale poorly and require significant computational resources. An approach based on systematic transformations of the problem under study to a form that can be addressed with simpler methods is presented.

A method for computing a perfectly conducting periodic rectangular waveguide of the ladder type is proposed. A field transformation matrix is constructed that relates the complex amplitudes of the field in waveguide cross sections separated by the distance equal to the period of the system. The dispersion characteristics of a periodic waveguide operating in the terahertz range are calculated.

Monotonicity conditions for the CABARET scheme approximating a quasilinear scalar conservation law with a convex flux are obtained. It is shown that the monotonicity of the CABARET scheme for Courant numbers \(r \in (0.5,1]\) does not ensure the complete decay of unstable strong discontinuities. For the CABARET scheme, a difference analogue of an entropy inequality is derived and a method is proposed ensuring the complete decay of unstable strong discontinuities in the difference solution for any Courant number at which the CABARET scheme is stable. Test computations illustrating these properties of the CABARET scheme are presented.

Stable and unstable disturbances of ocean currents are studied by analyzing a spectral problem based on the evolution potential vorticity equation in the quasi-geostrophic approximation. The problem is reduced to a fourth-order nonself-adjoint differential equation with a small parameter multiplying the highest derivative and with several dimensionless physical parameters. A feature of the problem is that the spectral parameter is involved in both the equation and boundary conditions for the third derivative. The problem is considered in two versions, namely, with a boundary condition setting the function or its second derivative to zero. An efficient analytical-numerical method is constructed for solving the problem. According to this method, even and odd functions are computed using power series expansions of the solution at boundary and middle points of the layer. An equation for the desired spectrum of the problem is derived by matching the expansions at an interior point. The asymptotic expansions of solutions and eigenvalues for small values of the wave number \(k\) are studied. It is found that the problem for even and odd solutions with the boundary condition for the second derivative has a single finite eigenvalue and a countable set of indefinitely increasing eigenvalues as \(k \to 0\). The problem with the boundary condition for the function has only a countable set of indefinitely increasing eigenvalues as \(k \to 0\). The eigenvalues are computed for various parameters of the problem. The numerical results show that a current can be unstable in a wide range of \(k\).