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## On a finite element method for measure-valued optimal control problems governed by the 1D generalized wave equation

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.

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

Trautmann P., Vexler B., Zlotnik A., / Cornell University. Series "Working papers by Cornell University". 2017. No. 1702.00362.

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 in time bilinear ...

Added: February 2, 2017

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

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

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

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 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., 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., Mathematical Modelling and Analysis 2021 Vol. 26 No. 3 P. 479-502

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 alternative ...

Added: December 9, 2020

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

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., 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 A., Čiegis R., / 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

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 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

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

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

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

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

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., 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.A., Zlotnik I. A., Доклады Академии наук 2012 Vol. 86 No. 3 P. 750-755

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: October 4, 2012

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.A., Chetverushkin B. N., Differential Equations 2020 Vol. 56 No. 7 P. 910-922

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized at a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study an implicit three-level weighted difference scheme and an implicit two-level vector difference scheme. The important domination property of the operator of ...

Added: July 16, 2020