On the relation between the entropy balance and the numerical solutions of systems of conservation laws

This work deals with the relation between the numerical solutions of hyperbolic systems of conservation laws and the associated entropy evolution. An analysis of the continuum problem by means of variational calculus clearly emphasizes the consequences of the adopted reconstruction procedure on the induced entropy balance. A methodology is proposed that allows for a posterior local and global spurious entropy production estimates on the basis of an additional equation representing a discrete approximation to the entropy inequality. The problem of defining a consistent approximation of the numerical entropy flux is also addressed in detail. Properly designed numerical experiments support the analysis and contribute to providing a more comprehensive evaluation of the numerical entropy dynamics. © 1997 John Wiley & Sons, Ltd.     

On integration of constraints imposed on a system of conservation laws by the second law of thermodynamics

In this paper the explicit form of such systems of conservation laws, which imply an additional conservation law, is determined. It is shown that there are many classes of such systems besides the symmetric conservative ones. From the point of view of phenomenological thermodynamics, this result can be interpreted as a sort of integration of the constraints imposed by the second law of thermodynamics.     

Properties of thermocapillary fluids and symmetrization of motion equations

The equations of fluid motions are considered in the case of internal energy depending on mass density, volume entropy and their spatial derivatives. The model corresponds to domains with large density gradients in which the temperature is not necessarily uniform. The new general representation is written in symmetric form with respect to the mass and entropy densities. For conservative motions of perfect thermocapillary fluids, Kelvin's circulation theorems are always valid. Dissipative cases are also considered; we obtain the balance of energy and we prove that equations are compatible with the second law of thermodynamics. The internal energy form allows to obtain a Legendre transformation inducing a quasi-linear system of conservation laws which can be written in a divergence form and the stability near equilibrium positions can be deduced. The result extends classical hyperbolicity theory for governing-equations' systems in hydrodynamics, but symmetric matrices are replaced by Hermitian matrices.     

Non-equilibrium relativistic M.H.D. with finite electrical conductivity

A system of balance laws for relativistic m.h.d, with finite eIectrical conductivity, heat flux and viscosity is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance). Adopting a two-fluid scheme the plasma is treated as a mixture of a neutral fluid and a charged fluid. Following the approach ofextended thermodynamics heat flux, viscous stress and electric current density are considered as new field variables contributing to non equilibrium entropy density and flux.     

求解双曲守恒律方程的WENO型熵相容格式

通过在单元交界面处进行高阶WENO重构,得到了一种求解双曲型守恒律方程的WENO型熵相容格式。用该格式对一维Burgers方程和Euler方程进行数值模拟,结果表明,该格式具有高精度、基本无振荡性等特点。     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

Approximate conservation laws of perturbed partial differential equations

This paper presents a general result on approximate conservation laws of perturbed partial differential equations. A method of constructing approximate conservation laws to systems of perturbed partial differential equations is given, which is based on approximate Noether symmetries of approximate and standard adjoint systems of the original system. The relationship between the Noether symmetry operators of approximate and standard adjoint system is established. As a result, the approach is applied to the perturbed wave equation and the perturbed KdV equation.     

Extended thermodynamics of a relativistic plasma with finite electric conductivity

A system of equations for relativistic m.h.d, with finite electric conductivity and no heat flux is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance) with convex density (symmetric-hyperbolic systems). The electric current density is treated as a new field variable which contributes to non equilibrium entropy density (extended thermodynamics). The result is a theory in which only one new constitutive function, representing entropy increment respect to equilibrium, is necessary to characterize the properties of the medium related to electric conductivity.G.N.F.M. of the C.N.R.     

Entropy and Global Existence for Hyperbolic Balance Laws

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchuts discrete velocity BGK models approximating hyperbolic systems of conservation laws.     

A Material Momentum Balance Law for Rods

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin’s circulation theorem. A novel identity for the singular sources at a discontinuity is also established. Dedicated to James N. Flavin, my friend and mentor, on the occasion of his 70th Birthday.     

Moment equations in the kinetic theory of gases and wave velocities

We consider the evolution system for N-moments of the Boltzmann equation and we require the compatibility with an entropy law. This implies that the distribution function depends only on a single scalar variable which is a polynomial in . It is then possible to construct the generators such that the system assumes a symmetric hyperbolic form in the main field. For an arbitrary we prove that the systems obtained maximise the entropy density. If we require that the entropy coincides with the usual one of non-degenerate gases, we obtain an exponential function for , which was already found by Dreyer. From these results the behaviour of the characteristic wave velocities for an increasing number of moments is studied and we show that in the classical theory the maximum velocity increases and tends to infinity, while in the relativistic case the wave and shock velocities are bounded by the speed of light. Received June 5, 1997     

Method of <Emphasis Type="Italic">A</Emphasis>-operators and conservation laws for the equations of gas dynamics

An algorithm is proposed which allows all conservation laws for a system of differential equations to be to obtained from its one zero-order conservation law for which the general rank of the Jacobi matrix is equal to the number of independent variables of the system. The efficiency of the algorithm is shown by examples of the equations of gas dynamics, for which new conservation laws are derived. For the equations considered, additional symmetry properties related to these conservation laws are established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 53–60, March–April, 2009.     

Higher-order jump conditions for conservation laws

The hyperbolic conservation laws admit discontinuous solutions where the solution variables can have finite jumps in space and time. The jump conditions for conservation laws are expressed in terms of the speed of the discontinuity and the state variables on both sides. An example from the Gas Dynamics is the Rankine–Hugoniot conditions for the shock speed. Here, we provide an expression for the acceleration of the discontinuity in terms of the state variables and their spatial derivatives on both sides. We derive a jump condition for the shock acceleration. Using this general expression, we show how to obtain explicit shock acceleration formulas for nonlinear hyperbolic conservation laws. We start with the Burgers’ equation and check the derived formula with an analytical solution. We next derive formulas for the Shallow Water Equations and the Euler Equations of Gas Dynamics. We will verify our formulas for the Euler Equations using an exact solution for the spherically symmetric blast wave problem. In addition, we discuss the potential use of these formulas for the implementation of shock fitting methods.     

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

THE DERIVATION OF REYNOLDS TRANSPORT EQUATION FOR ARBITRARY MOTION CONTROL VOLUME BASED ON THE DYNAMIC BOUNDARY CALCULUS

严格而言,流体力学中所有守恒定律均是针对物质体系的(或称流体系统),如质量、动量、动量矩和能量等守恒定律。如果跟随物质体系描述和表征流体质点系的运动行为,即为Lagrange描述方法;如果把物质体系的运动和守恒定律转换到空间坐标系中,即为人们常说的Euler描述方法。因此,对于具体考察(跟随的)的流体物质系统而言,各守恒定律存在由物质体系表征到空间体系表征的转换,这个转换关系就是著名的Reynolds输运方程。本文从动边界微积分关系式出发,系统推导了在不同运动速度控制体上的雷诺输运方程,并通过讨论进一步阐明各种不同形式输运方程的物理意义。     

Stress intensity factor calculations based on a conservation integral

It is observed that one of the integral conservation laws of elastostatics, the so-called M-integral conservation law, has certain special features which make it possible to apply this conservation law for a class of plane elastic crack problems in order to calculate the elastic stress intensity factor in each case without solving the corresponding boundary value problem. The main characteristics which a problem must have in order for the approach to be useful are (1) for points very near to the origin of coordinates, the known elastic stresses are 0(r^?r) where r is the radial coordinate and γ ? 1, (2) for points very far from the origin, the known elastic stresses are 0(r^?r) where γ ? 1, and (3) the boundary of the body is made up of radial lines on which certain traction and/or displacement conditions are satisfied. The approach is demonstrated by determining the stress intensity factors for four familiar elastic crack problems directly from the conservation law, and then four similar additional applications of the M-integral conservation law are discussed.     

Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities

This paper obtains the conservation laws of the Klein–Gordon equation with power law and log law nonlinearities. The multiplier approach with Lie symmetry analysis is employed to obtain the conserved densities. The 1-soliton solutions are subsequently used to compute the conserved quantities from the conserved densities. Later the perturbation terms are added and the conservation laws of the perturbed Klein–Gordon equation are studied.     

Integrating factors and conservation laws for nonconservative dynamical systems

A general approach to the construction of conservation laws for classical nonconservative dynamical systems is presented. The conservation laws are constructed by finding corresponding integrating factors for the equations of motion. Necessary conditions for existence of the conservation laws are studied in detail. A connection between an a priori known conservation law and the corresponding integrating factors is established. The theory is applied to two particular problems.