Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier–Stokes equations of a viscous compressible heat-conducting gas.     

On new spatial discretization of the multidimensional quasi-gasdynamic system of equations with nondecreasing total entropy

The multidimensional quasi-gasdynamic system of equations written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization on a nonuniform rectangular grid is constructed for this system. The basic unknown functions (density, velocity, and temperature) are defined on a common grid, while the fluxes and viscous stresses, on staggered grids. The discretization is specially constructed so that the total entropy does not decrease, which is achieved by applying numerous original features.     

Properties of an Aggregated Quasi-Gasdynamic System of Equations for a Homogeneous Gas Mixture

Doklady Mathematics - For an aggregated quasi-gasdynamic system of equations for a homogeneous gas mixture, we give an entropy balance equation with a nonnegative entropy production in the presence...     

On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force

A multidimensional barotropic quasi-gasdynamic system of equations in the form of mass and momentum conservation laws with a general gas equation of state p = p(ρ) with p′(ρ) > 0 and a potential body force is considered. For this system, two new symmetric spatial discretizations on nonuniform rectangular grids are constructed (in which the density and velocity are defined on the basic grid, while the components of the regularized mass flux and the viscous stress tensor are defined on staggered grids). These discretizations involve nonstandard approximations for ?p(ρ), div(ρu), and ρ. As a result, a discrete total mass conservation law and a discrete energy inequality guaranteeing that the total energy does not grow with time can be derived. Importantly, these discretizations have the additional property of being well-balanced for equilibrium solutions. Another conservative discretization is discussed in which all mass flux components and viscous stresses are defined on the same grid. For the simpler barotropic quasi-hydrodynamic system of equations, the corresponding simplifications of the constructed discretizations have similar properties.     

Entropy balance for the one-dimensional hyperbolic quasi-gasdynamic system of equations

Entropy balance in the one-dimensional hyperbolic quasi-gasdynamic (HQGD) system of equations is analyzed. In particular, in regular flow regimes, it is shown that the behavior of entropy in the HQGD system is mainly determined by terms involving the natural viscosity and thermal conductivity coefficients. The total entropy production differs from the Navier–Stokes equations for viscous compressible heat-conducting gases by O(τ₂) terms, where τ is a relaxation parameter. Additionally, a similar analysis of energy balance is performed for the simpler case of the barotropic HQGD system, which is of interest for some applications.

     

Conditions for Dissipativity of an Explicit Finite-Difference Scheme for a Linearized Multidimensional Quasi-Gasdynamic System of Equations

Doklady Mathematics - We study an explicit two-level finite-difference scheme for a linearized multidimensional quasi-gasdynamic system of equations. For an initial-boundary value problem on a...     

Analysis of a set of nonlinear dynamics trajectories: Stability of difference equations

For a set of difference equations generated by discretization of the set of differential equations with Hukuhara derivative a principle of comparison with matrix Lyapunov function is specified and sufficient stability conditions of certain type are established. The analysis is carried out in terms of a matrix Lyapunov function of special structure. For an essentially nonlinear multiconnected switched difference system, conditions are obtained providing the asymptotic stability of its zero solution for any switching law. An example is presented to demonstrate efficiency of the proposed approaches.     

Vacuum particle creation under action of a strong external field: an example of irreversible behavior of a system with time reversal symmetry

On the basis of a kinetic equation (KE) of the Vlasov type, which is a strong non-perturbative consequence of the QED equations of motion in the quasiparticle representation, we investigate the irreversible behavior of the electron-positron plasma (EPP) created from vacuum under action of a pulsed external quasiclassical field. The basic KE is upon time reversal. Nevertheless the EPP reveals some features characteristic for irreversible processes. The nondecreasing entropy principle is introduced for the asymptotic in- and out-states: S _out ?? S _in . Some features of the entropy evolution are investigated for different excitation levels of the EPP.     

Weak-strong uniqueness for the compressible Navier-Stokes equations with a hard-sphere pressure law

We consider the Navier-Stokes equations with a pressure function satisfying a hard-sphere law. That means the pressure, as a function of the density, becomes infinite when the density approaches a finite critical value. Under some structural constraints imposed on the pressure law, we show a weak-strong uniqueness principle in periodic spatial domains. The method is based on a modified relative entropy inequality for the system. The main difficulty is that the pressure potential associated with the internal energy of the system is largely dominated by the pressure itself in the area close to the critical density. As a result, several terms appearing in the relative energy inequality cannot be controlled by the total energy.     

Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D

Summary. Systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms. To solve these weakly coupled systems numerically a class of explicit and implicit upwind finite volume methods on unstructured grids is presented. Provided an unique entropy solution of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing discretization parameter to this entropy solution. These results are applied to examples from combustion theory and hydrology where the existence of entropy solutions can be shown. The proofs rely on an extension of a result due to DiPerna concerning measure valued solutions to the case of weakly coupled hyperbolic systems. Received April 29, 1997     

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W_j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W_j converge to 0 exponentially quickly for large enough j. If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.     

Energy equalities and estimates for barotropic quasi-gasdynamic and quasi-hydrodynamic systems of equations

Energy equalities are derived for the barotropic quasi-gasdynamic, modified quasi-gasdynamic, and quasi-hydrodynamic systems of equations. Global energy estimates of solutions are obtained. For the second of the systems, necessary and sufficient conditions for nonuniform and uniform Petrovskii parabolicity are derived.     

A numerical method for the Cahn–Hilliard equation with a variable mobility

We consider a conservative nonlinear multigrid method for the Cahn–Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank–Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem has the conservation of mass and the decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.     

Thermal effects in the description of a multicomponent mixture using the density functional method

The use of a model, based on an expression for the total entropy in the form of a functional with the temperature and density gradients of the components, is proposed to describe a multicomponent, multiphase system using continuous hydrodynamics (that is, within the framework of the approach of the continuum mechanics without discontinuities in the hydrodynamic quantities). It is proved that this model is consistent with the zeroth law of thermodynamics. Expressions for the stress tensor, the diffusion fluxes and the heat flux are found from the condition that the entropy production is non-negative. Compared with the classical Newton, Fick and Fourier laws, these expressions contain third-order spatial derivatives, The problem of a mixture between two parallel and impermeable walls at different temperatures is analysed. In this case, the system of dynamic equations reduces to a system of ordinary differential equations. It is shown that the number of free parameters, on which the solution depends, corresponds to the number of boundary and general integral conditions.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).     

On the entropy conditions for some flux limited diffusion equations

In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.     

Necessity of entropy control in gasdynamic computations

A recently published numerical approach is used to perform an analytical study of the behavior of entropy in the numerical integration of gas dynamics equations. The study confirms the existence of discontinuous solutions for which the principle of nondecreasing entropy can be violated. A simple entropy-controlling algorithm is proposed for gasdynamic flow computations and other numerical methods.     

Artificial Dissipation Coefficients in Regularized Equations of Supersonic Aerodynamics

Doklady Mathematics - A method for introducing artificial dissipation coefficients into a numerical algorithm based on the quasi-gasdynamic system of equations is proposed. The method applies to...     

One-step and extrapolation methods for differential-algebraic systems

Summary The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.