KRM
Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation
Alexander Zlotnik Ilya Zlotnik
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 time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.
keywords: approximate and discrete transparent boundary conditions The time-dependent Schrödinger equation the tunnel effect. stability finite element method discrete convolution operator Galerkin method
KRM
The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis
Alexander Zlotnik
We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The finite-difference 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 prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well.
keywords: error estimates approximate transparent boundary conditions Time-dependent Schrödinger equation stability Crank-Nicolson scheme Numerov scheme unbounded domain global Richardson extrapolation.
KRM
On a regularization of the magnetic gas dynamics system of equations
Bernard Ducomet Alexander Zlotnik
A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant magnetic regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.
keywords: regularization Magnetic gas dynamics entropy balance equation.
KRM
On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation
Bernard Ducomet Alexander Zlotnik Ilya Zlotnik
We consider a 1D Schrödinger equation with variable coefficients on the half-axis. We study a family of two-level symmetric finite-difference schemes with a three-point parameter dependent averaging in space. This family includes a number of particular schemes. The schemes are coupled to an approximate transparent boundary condition (TBC). We prove two stability bounds with respect to initial data and a free term in the main equation, under suitable conditions on an operator of the approximate TBC. We also consider the family of schemes on an infinite mesh in space. We derive and analyze the discrete TBC allowing to restrict these schemes to a finite mesh and prove the stability conditions for it. Numerical examples are also included.
keywords: finite-difference schemes the time-dependent Schrödinger equation finite element method approximate and discrete transparent boundary conditions Stability discrete convolution operator.
MCRF
Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients
Philip Trautmann Boris Vexler Alexander Zlotnik

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$ or vector measures $\mathcal M({\mathit \Omega }, L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $α\|u\|_{\mathcal M_T}$ with $α>0$. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

keywords: Wave equation optimal control measure-valued control vector measure control finite element method stability error estimates

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