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Семейство разностных схем с прозрачными граничными условиями для нестационарного уравнения Шрёдингера в полуполосе
Доклады Академии наук. 2011. Т. 436. № 1. С. 19-25.
Zlotnik A., Zlotnik I. A.
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.
Keywords: устойчивостьtwo-dimensional nonstationary Schrodinger equationtwo-level symmetric finite-difference schemes with averagesdiscrete transparent boundary conditionsstabilityimplementationдвумерное нестационарное уравнение Шрёдингерадвухслойные симметричные разностные схемы с усреднениямидискретные прозрачные граничные условияреализация
Zlotnik A.A., Zlotnik I.A., Доклады Академии наук 2011 Vol. 83 No. 1 P. 12-18
В работе решается начально-краевая задача для обобщенного уравнения Шрёдингера (с переменными коэффициентами) в полуполосе; она возникает, например, в некоторых задачах ядерной физики.
Строится новое семейство двухслойных симметричных разностных схем с усреднениями по пространственным переменным на конечной сетке, охватывающее набор различных по способу построения схем. Рассмотренное с абстрактным прозрачным граничным условием (ПГУ), оно является абсолютно устойчивым в ...
Added: December 22, 2015
Ducomet Bernard, Zlotnik Alexander, Romanova Alla, Applied Mathematics and Computation 2015 Vol. 255 P. 195-206
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 ...
Added: October 10, 2014
Zlotnik A., Romanova A., Applied Numerical Mathematics 2015 Vol. 93 P. 279-294
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time $L^2$-stability (in particular, $L^2$-conservativeness) together ...
Added: November 30, 2013
Zlotnik A., Koltsova N., / Cornell University. Series math "arxiv.org". 2012. No. arXiv:1211.3613 [math.NA].
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 averagings 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 ...
Added: January 25, 2013
Zlotnik A., Koltsova N., Computational Methods in Applied Mathematics 2013 Vol. 13 No. 2 P. 119-138
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 ...
Added: April 6, 2013
Zlotnik Alexander, / Cornell University. Series math "arxiv.org". 2015.
We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger 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 transpa\-rent boundary conditions is considered. We present its stability properties and derive new error ...
Added: March 27, 2015
Ducomet B., Zlotnik A., Romanova A. V., / Cornell University. Series math "arxiv.org". 2013. No. 1309.7280.
An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite (or 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 ...
Added: October 1, 2013
Zlotnik A., Romanova A. V., / Cornell University. Series math "arxiv.org". 2013. No. arxiv: 1307.5398.
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L_2-stability (in particular, L_2-conservativeness) are ...
Added: July 24, 2013
Zlotnik Alexander, Zlotnik Ilya, Computational Methods in Applied Mathematics 2015 Vol. 15 No. 2 P. 233-245
We consider the Cauchy problem for the 1D generalized Schrὅdinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order ...
Added: March 3, 2015
Trautmann P., Vexler B., Zlotnik A., Mathematical Control and Related Fields 2018 Vol. 8 No. 2 P. 411-449
This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level ...
Added: April 8, 2017
Zlotnik A., Kireeva O., / Cornell University. Series arXiv "math". 2020. No. arXiv:2011.14104v2[math.NA].
We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The ...
Added: December 1, 2020
А. А. Злотник, Т. А. Ломоносов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2020 Т. 492 № 1 С. 31-37
We study an explicit two-level symmetric in space finite-difference scheme for the multi\-di\-men\-si\-onal barotropic gas dynamics system of equations with quasi-gasdynamic regulari\-za\-tion linearized at a constant solution (with arbitrary velocity). A criterion and both necessary and sufficient conditions for the $L^2$-dissipativity of the solutions to the Cauchy problem for the scheme are derived by the spectral ...
Added: March 4, 2020
Zlotnik A., Lomonosov T., Applied Mathematics Letters 2020 Vol. 103 Article 106198
We study an explicit in time and symmetric in space finite-difference scheme with a kinetic regularization for the 2D and 3D gas dynamics system of equations linearized at a constant solution (with any velocity). We derive both necessary and sufficient conditions for $L^2$-dissipativity of the Cauchy problem for the scheme by the spectral method. The Courant number ...
Added: December 21, 2019
Zlotnik A., Кольцова Н. В., Вестник Московского энергетического института 2012 № 6 С. 72-94
Рассмотрено решение начально-краевой задачи для одномерного самосопряженного параболического уравнения на полуоси. Изучено широкое семейство двухслойных разностных схем с двумя параметрами — с усреднениями с весами как по времени, так и по пространству. Доказана их устойчивость в двух нормах энергетическим методом. Строго выведены дискретные прозрачные граничные условия методом производящих функций. Приведены результаты численных экспериментов.
Работа выполнена при ...
Added: January 23, 2013
СПб. : Издательство Санкт-Петербургского университета, 2008
В сборнике представлены результаты исследований по механике сплошной среды, в основном задач колебаний и устойчивости упругих конструкций. Характерной чертой исследований является использование разнообразных компьютерных методов: методов вычислительной механики сплошной среды, компьютерной алгебры, визуализации и др. Анализ опирается на сопоставление данных, полученных в различных подходах, причем наиболее часто сопоставляются результаты, полученные асимптотическими методами и по методу ...
Added: February 4, 2013
Zlotnik Alexander, Ducomet Bernard, Zlotnik Ilya et al., , in : Numerical Mathematics and Advanced Applications - ENUMATH 2013. Vol. 103.: Springer, 2015. P. 203-211.
The time-dependent Schrödinger equation is the key one in many fields. It should be often solved in unbounded space domains. Several approaches are known to deal with such problems using approximate transparent boundary conditions (TBCs) on the artificial boundaries. Among them, there exist the so-called discrete TBCs whose advantages are the complete absence of spurious ...
Added: October 10, 2014
Zlotnik A., , in : Finite Difference Methods, Theory and Applications 6th International Conference, FDM 2014, Lozenetz, Bulgaria, June 18-23, 2014, Revised Selected Papers. Vol. 9045.: Zürich : Springer, 2015. P. 129-141.
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 ...
Added: March 23, 2015
Vexler B., Zlotnik A., Trautmann P., Comptes Rendus Mathematique 2018 Vol. 356 No. 5 P. 523-531
The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions or vector measures. Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished. ...
Added: April 8, 2017
Ducomet B., Zlotnik A., Zlotnik I. A., / Cornell University. Series math "arxiv.org". 2013. No. arxiv: 1303.3471.
We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Crank-Nicolson finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniform in time L2-stability is proved. Due to the ...
Added: March 16, 2013
Zlotnik A., Applied Mathematics Letters 2019 Vol. 92 P. 115-120
We deal with an explicit finite-difference scheme with a regularization for the 1D gas dynamics equations linearized at the constant solution. The sufficient condition on the Courant number for the $L^2$-dissipativity of the scheme is derived in the case of the Cauchy problem and a non-uniform spatial mesh. The energy-type technique is developed to this end, and ...
Added: January 20, 2019
Burov A. A., Герман А. Д., Косенко И. И. et al., Acta Astronautica 2018 Vol. 143 P. 126-132
Relative equilibria of a pendulum attached to the surface of a uniformly rotating celestial body are considered. The locations of the tether anchor that correspond to a given spacecraft position are defined. The domains, where the spacecraft can be held with the help of such a pendulum, are also described. Stability of the found relative ...
Added: September 10, 2018
Zlotnik Alexander, Zlotnik Ilya, / Cornell University. Series math "arxiv.org". 2014. No. arXiv:1405.3147.
We consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the Richardson extrapolation to improve significantly the accuracy in time step. ...
Added: May 14, 2014
Ducomet Bernard, Zlotnik Alexander, Zlotnik Ilya, ESAIM: Mathematical Modelling and Numerical Analysis 2014 Vol. 48 No. 6 P. 1681-1699
We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. ...
Added: May 23, 2014
Ruderman M. S., Petrukhin N., Pelinovsky E., Solar Physics 2021 Vol. 296 Article 124
In this article we study the plasma motion in the transitional layer of a coronal loop randomly
driven at one of its footpoints in the thin-tube and thin-boundary-layer (TTTB) approximation. We introduce the average of the square of a random function with respect to time. This
average can be considered as the square of the oscillation amplitude ...
Added: August 18, 2021