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## Verification of an Entropy Dissipative QGD-Scheme for the 1D Gas Dynamics Equations

An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh $L^1$-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for $L^2$-dissipativity of the Cauchy problem for a linearized QGD-scheme.

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 method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common 2nd order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain \textit{much more} accurate results using much less amount of both elements and time steps.

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.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

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 with the error estimate $O(\tau^2+|h|^4)$ are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip and solving of tridiagonal systems in its main direction is developed to implement the splitting method for general potential. We also engage the Richardson extrapolation in time to increase the error order with respect to time step and get the method of higher order both in space and time. Numerical results on the tunnel effect for smooth and discontinuous rectangular barriers are included together with the careful practical error analysis on refining meshes.

We consider explicit two-level three-point in space finite-difference schemes for solving 1D barotropic gas dynamics equations. The schemes are based on special quasi-gasdynamic and quasi-hydrodynamic regularizations of the system. We linearize the schemes on a constant solution and derive the von Neumann type necessary condition and a CFL type criterion (necessary and sufficient condition) for weak conservativeness in $L^2$ for the corresponding initial-value problem on the whole line. The criterion is essentially narrower than the necessary condition and wider than a sufficient one obtained recently in a particular case; moreover, it corresponds most well to numerical results for the original gas dynamics system.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.