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## On compact 4th order finite-difference schemes for the wave equation

math.
arXiv.
Cornell University
,
2020.
No. arXiv:2011.14104v2[math.NA].

Zlotnik A., Kireeva O.

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 technique is applicable to other types of PDEs including parabolic and time-dependent Schr\"{o}dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for $n\geq 2$. For $n=1$ and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Research target:
Mathematics

Language:
English

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., Romanova A. V., A Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip / 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., 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

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

Zlotnik A., Zlotnik I. A., 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., Romanova A. V., 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., 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., Čiegis R., Applied Mathematics Letters 2021 Vol. 115 Article 106949

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove ...

Added: December 9, 2020

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., Zlotnik I. A., 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 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., 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

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

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

Ducomet B., Zlotnik A., Zlotnik I. A., 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., 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

Ducomet B., Zlotnik A., Romanova A. V., 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., Lomonosov T., Журнал вычислительной математики и математической физики 2019 Т. 59 № 3 С. 481-493

Изучаются явные двухслойные по времени и симметричные по пространству разностные схемы, построенные посредством аппроксимации 1D баротропных квазигазо/квазигидродинамических систем уравнений. Они линеаризуются на постоянном решении с ненулевой скоростью, и для них выводятся как необходимые, так и достаточные условия $L^2$-диссипативности решений задачи Коши в зависимости от числа Маха. Эти условия различаются между собой не более чем в 2 раза. Результаты ...

Added: September 26, 2018

Zlotnik A., Lomonosov T., 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., 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., 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

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 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