L^2-диссипативность разностных схем для регуляризованных 1D баротропных уравнений движения газа при малых числах Маха
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.
We consider the Cahn-Hilliard equation in one space dimension with scaling parameter epsilon, i.e., u(t) = (W'(u) - epsilon(2)u(xx))(xx), where W is a nonconvex potential. In the limit epsilon down arrow 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.
A study of a regularized quasi-hydrodynamic system of equations for the two-component isothermal mixture with the diffuse interface is accomplished. Under general assumptions on the Helmholtz free energy of the mixture, the energy balance equation with non-positive energy production and its corollary, the law of non-increasing total energy, are derived. The necessary and sufficient conditions for the linearized stability of constant solutions are derived (in a particular case). A finite-difference approximation of the problem is constructed in the 2D periodic case for a non-uniform rectangular mesh. The results of numerical experiments are presented which demonstrate the qualitative correctness of the model and applicability of the linearized stability criterion in the original nonlinear statement.
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.
We study an explicit two-level in time and three-point symmetric in space finite-difference scheme for 1D barotropic and full gas dynamics systems of equations. The scheme is a linearization at a constant background solution (with an arbitrary velocity) of finite-difference schemes with general viscous regularization. We enlarge recently proved sufficient conditions (on the Courant-like number) for $L^2$-dissipativity in the Cauchy problem for the schemes by deriving new bounds for the commutator of matrices of viscous and convective terms. We deal with the case of a kinetic regularization in more detail and specify sufficient conditions in this case where the mentioned matrices are closely connected. Importantly, these new sufficient conditions rapidly tend to the known necessary ones as the Mach number grows. Also several forms of setting a regularization parameter are considered.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.
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.