### Book chapter

## Методы с расщеплением по потенциалу и дискретными ПГУ для численного решения нестационарного уравнения Шрёдингера в неограниченном параллелепипеде

### In book

This volume is dedicated to the 80th anniversary of academician V. M. Matrosov. The book contains reviews and original articles, which address the issues of development of the method of vector Lyapunov functions, questions of stability and stabilization control in mechanical systems, stability in differential games, the study of systems with multirate time and other. Articles prepared specially for this edition.

An initial-boundary value problem for the *n *-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite parallelepiped is considered. Starting from the Numerov–Crank–Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting double-splitting method, the uniqueness of solution and the uniform in time L2-stability are proved and an error estimate is stated. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applied to implement the scheme for general potential.

Proceedings of the III International Conference in memory of V.I. Zubov "Stability and Control Processes (SCP 2015)".

Data mining aims at finding interesting patterns from datasets, where “interesting” means reflecting intrinsic dependencies in the domain of interest rather than just in the dataset. Concept stability is a popular relevancy measure in FCA but its behaviour have never been studied on various datasets. In this paper we propose an approach to study this behaviour. Our approach is based on a comparison of stability computation on datasets produced by the same general population. Experimental results of this paper show that high stability of a concept in one dataset suggests that concepts with the same intent in other dataset drawn from the population have also high stability. Moreover, experiments shows some asymptotic behaviour of stability in such kind of experiments when dataset size increases.

We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded $n$-dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error estimates $O(\tau^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averages both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.

We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S|/ + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/. We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.

An initial–boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved.

A new family of two level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed.