### ?

## A Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

arxiv.org.
math.
Cornell University
,
2013.
No. arxiv: 1307.5398.

Zlotnik A., Romanova A. V.

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 proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip is developed to implement the splitting method for general potential. Numerical results on the tunnel effect for smooth and rectangular barriers together with the practical error analysis on refining meshes are included as well.

Language:
English

Keywords: устойчивостьdiscrete transparent boundary conditionsstabilityтуннельный эффектединственностьuniquenessнестационарное уравнение ШрёдингераThe Strang splittingрасщепление Стренгаtunnel effectthe time-dependent Schrodinger equationthe Numerov discretization in spacethe Crank-Nicolson discretization in timepractical error analysisдискретизация Нумерова по пространствудискретизация Кранк-Никольсон по временипрактический анализ погрешности

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

Ducomet B., Zlotnik A., Romanova A. V., A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped / 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 Alexander, On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped / 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

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

Ducomet B., Zlotnik A., Zlotnik I. A., The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip / 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

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

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

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

Zlotnik A., В кн. : Актуальные проблемы математической физики. Сборник тезисов докладов. : М. : Физический факультет МГУ им. М.В. Ломоносова, 2014. С. 48-51.

Уравнение Шрёдингера играет важную роль в квантовой механике и электронике, ядерной, атомной, волновой физике и др. Часто его необходимо решать в неограниченных областях. Для этой цели разработан ряд методов, обычно использующих приближенные прозрачные граничные условия (ПГУ) на искусственных границах, в том числе дискретные ПГУ. Для последних полностью отсутствуют отражения от искусственных границ на практике и ...

Added: November 28, 2014

Zlotnik Alexander, Kinetic and Related Models 2015 Vol. 8 No. 3 P. 587-613

We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we ...

Added: November 27, 2014

Zlotnik A., Zlotnik I. A., Kinetic and Related Models 2012 Vol. 5 No. 3 P. 639-667

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large x. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in ...

Added: March 21, 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 A., Kireeva O., On compact 4th order finite-difference schemes for the wave equation / 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

Zlotnik Alexander, Zlotnik Ilya, On the Richardson extrapolation in time of finite element method with discrete TBCs for the Cauchy problem for the 1D Schrödinger equation / 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

Zlotnik Alexander, Zlotnik Ilya, Some remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis / Cornell University. Series math "arxiv.org". 2014. No. arXiv.org:1406.5102.

We consider the generalized time-dependent Schr\"odinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step $h$. Next, for a selected scheme of the family, we discover that the discrete convolution in time in ...

Added: June 20, 2014

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 A., Kireeva O., Mathematical Modelling and Analysis 2018 Vol. 23 No. 3 P. 359-378

We deal with the standard three-level bilinear FEM and finite-difference scheme to solve the initial-boundary value problem for the 1D wave equation. We consider initial data and the free term which are the Dirac delta-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the $L^2$, $L^1$, ...

Added: January 14, 2018

Zlotnik A., Lapukhina A. V., Journal of Mathematical Sciences 2010 Vol. 169 No. 1 P. 84-97

We consider an initial-boundary value problem for the one-dimensional nonstationary Schrödinger equation on the half-axis and study a two-level symmetric finite-difference scheme of Numerov type with higher approximation order. This scheme is constructed on a finite mesh, which is uniform with respect to space, with a nonlocal approximate transparent boundary condition of a general form ...

Added: December 23, 2015

Злотник А.А., Лапухина А., Проблемы математического анализа 2010 № 47 С. 77-88

Нестационарное уравнение Шрёдингера относится к основным уравнениям математической физики и находит многочисленные приложения. Очень часто его приходится численно решать в неограниченных по пространству областях. Для этой цели разработан ряд подходов, связанных с постановкой искусственных или приближенных прозрачных граничных условий (ПГУ) на искусственных границах. Среди них следует выделить подход, использующий так называемые дискретные ПГУ. Серьезный практический ...

Added: December 22, 2015

Zlotnik A., Čiegis R., On properties of compact 4th order finite-difference schemes for the variable coefficient wave equation / Cornell University. Series arXiv "math". 2021. No. ArXiv: 2101.10575v2[math.NA].

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some ...

Added: February 2, 2021

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., Koltsova N., On a family of finite-difference schemes with discrete transparent boundary conditions for a parabolic equation on the half-axis / 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., Zlotnik I. A., Доклады Академии наук 2011 Т. 436 № 1 С. 19-25

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 ...

Added: July 5, 2012

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