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## Conditions for L2-Dissipativity of Linearized Explicit Difference Schemes with Regularization for 1D Barotropic Gas Dynamics Equations

Explicit two-level in time and symmetric in space finite-difference schemes constructed by approximating the 1D barotropic quasi-gas-/quasi-hydrodynamic systems of equations are studied. The schemes are linearized about a constant solution with a nonzero velocity, and, for them, necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived depending on the Mach number. These conditions differ from one another by at most twice. The results substantially develop the ones known for the linearized Lax–Wendroff scheme. Numerical

experiments are performed to analyze the applicability of the found conditions in the nonlinear formulation

to several schemes for different Mach numbers.

We study explicit two-level finite-difference schemes on staggered meshes for two known regularizations of 1D barotropic gas dynamics equations including schemes with discretizations in *x *that possess the dissipativity property with respect to the total energy. We derive criterions of L^2-dissipativity in the Cauchy problem for their linearizations at a constant solution with zero background velocity. We compare the criterions for schemes on non-staggered and staggered meshes. Also we consider the case of 1D Navier-Stokes equations without artificial viscosity coefficient. To analyze the case of the 1D Navier-Stokes-Cahn-Hilliard equations, we derive and verify the criterions for L^2-dissipativity and stability for an explicit finite-difference scheme approximating a non-stationary 4th-order in *x *equation that includes a 2nd-order term in *x*. The obtained criteria may be useful to compute flows at small Mach numbers.

Key words: -dissipativity, explicit finite-difference schemes, staggered meshes, gas dynamics equations, Navier-Stokes-Cahn-Hilliard equations.

A scalable method for mining graph patterns stable under subsampling is proposed. The existing subsample stability and robustness measures are not antimonotonic according to definitions known so far. We study a broader notion of antimonotonicity for graph patterns, so that measures of subsample stability become antimonotonic. Then we propose gSOFIA for mining the most subsample-stable graph patterns. The experiments on numerous graph datasets show that gSOFIA is very efficient for discovering subsample-stable graph patterns.

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 formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed.

The object of study of this paper is a regional economic system which is complex, dynamic and developable by nature. The reproduction of material wealth necessary for the region is provided in the process of functioning of the above system through the interaction between the combinations of subjective (personal) and objective (material) elements, thereby meeting regional environmental and economic needs.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.