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On L^2-dissipativity of linearized explicit finite-difference schemes with a regularization on a non-uniform spatial mesh for the 1D gas dynamics equations
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 the proof is both short and under clear conditions on matrices of the convective and regularizing (dissipative) terms. A scheme with a kinetically motivated regularization is considered as an application in more detail.
Research target:
Mathematics
Language:
English
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., 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 Академии наук 2018 Т. 482 № 4 С. 375-380
Изучается явная двухслойная по времени и симметричная по пространству разностная схема, аппроксимирующая 1D квазигазодинамическую систему уравнений. Она линеаризуется на постоянном решении и для нее выводятся новые как необходимые, так и достаточные условия $L^2$-диссипативности решений задачи Коши, в том числе впервые при ненулевой фоновой скорости в зависимости от числа Маха.
Показано, что можно обеспечить независимость условия на число ...
Added: May 21, 2018
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
Lomonosov T., Journal of Mathematical Sciences 2020 Vol. 244 No. 4 P. 649-654
We obtain criteria for the L2-dissipativity of finite-difference schemes based on regularizations
of 1D barotropic and full gas dynamics systems of equations that are linearized
at a constant solution. Bibliography: 8 titles. ...
Added: February 18, 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., 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
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
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., Čiegis R., Applied Mathematics Letters 2018 Vol. 80 P. 35-40
The stability bounds and error estimates for a compact higher order Numerov-Crank-Nicolson scheme on non-uniform spatial meshes for the 1D time-dependent Schrödinger equation have been recently derived. This analysis has been done in $L^2$ and $H^1$ mesh norms and used the non-standard ``converse'' condition $h_\omega\leq c_0\tau$, where $h_\omega$ is the mean spatial step, $\tau$ is ...
Added: January 6, 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
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 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
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., 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
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.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
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., 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., 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., Lomonosov T., Doklady Mathematics 2018 Vol. 98 No. 2 P. 458-463
An explicit two-time-level and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2 -dissipativity of solutions of the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity ...
Added: September 12, 2018
Protasov V., Systems and Control Letters 2016 Vol. 90 P. 54-60
We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon ...
Added: February 22, 2016
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