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Showing papers on "Conservation law published in 1983"


Journal ArticleDOI
TL;DR: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes.
Abstract: This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is given to the Godunov-type schemes that result from using an approximate solution of the Riemann problem. For schemes based on flux splitting, the approximate Riemann solution can be interpreted as a solution of the collisionless Boltzmann equation.

3,133 citations


Journal ArticleDOI
TL;DR: It is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws using Godunov's and Roe's methods on a self-adjusting mesh.

790 citations


Journal ArticleDOI
TL;DR: In this article, an elementary but rigorous derivation for a variational principle for guiding center motion is given, and the application of variational principles in the derivation and solution of gyrokinetic equations is discussed.
Abstract: An elementary but rigorous derivation is given for a variational principle for guiding centre motion. The equations of motion resulting from the variational principle (the drift equations) possess exact conservation laws for phase volume, energy (for time-independent systems), and angular momentum (for azimuthally symmetric systems). The results of carrying the variational principle to higher order in the adiabatic parameter are displayed. The behaviour of guiding centre motion in azimuthally symmetric fields is discussed, and the role of angular momentum is clarified. The application of variational principles in the derivation and solution of gyrokinetic equations is discussed.

647 citations


Journal ArticleDOI
TL;DR: In this article, the stability and convergence of Glimm's random choice method applied to the Cauchy problem with initial data having small total variation was studied. But the convergence of the random choice was not investigated.
Abstract: Publisher Summary This chapter discusses a few results on the theoretical side of conservation laws concerning the convergence of approximate solutions. The general setting is a system of n conservation laws in one space dimension. In the context of conservation laws, the maximum norm and the total variation norm provide a natural pair of metrics in which to investigate stability; the maximum norm serving as a measure of the amplitude of the solution and the total variation norm as a measure of the gradient of the solution. Their role is indicated by the theorem concerning the stability and convergence of Glimm's random choice method applied to the Cauchy problem with initial data having small total variation.

544 citations


Journal ArticleDOI
TL;DR: Symmetric formulations in conservation form for the equations of gas dynamics are presented and the symmetrizability of systems of conservation laws which possess entropy functions is reviewed.

417 citations


Journal Article
TL;DR: In this article, Lagrangian formalism and conservation laws are combined with the idea of Gauge Invariance and non-Abelian Gauge Theories, including hidden symmetry and strong interactions among quarks.
Abstract: IntroductionLagrangian Formalism and Conservation LawsThe Idea of Gauge InvarianceNon-Abelian Gauge TheoriesHidden SymmetriesElectroweak Interactions of LeptonsElectroweak Interactions of QuarksStrong Interactions among QuarksUnified TheoriesEpilogue

341 citations


Book ChapterDOI
01 Jan 1983
TL;DR: In this article, the main difficulty in solving nonlinear partial differential equations lies in the following fact: after introducing a suitable sequence of approximations one needs enough a priori estimates to ensure the convergence of a subsequence to a solution; this argument is based on compactness results and in a nonlinear case one needs more estimates than in the linear case where weak continuity results can be used.
Abstract: One of the main difficulties in solving nonlinear partial differential equations lies in the following fact: after introducing a suitable sequence of approximations one needs enough a priori estimates to ensure the convergence of a subsequence to a solution; this argument is based on compactness results and in a nonlinear case one needs more estimates than in the linear case where weak continuity results can be used.

335 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws.
Abstract: Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatogra- phy, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coinci- dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of 2 X 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide. Introduction. Systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, nonlinear wave motion in elastic strings, as well as in multicomponent chromatography (1,4,5,6,9,11,12). These systems have many interesting features. The Riemann problem for these equations can be explicitly solved in the large, and wave interactions have a simplified structure, even in the presence of a nonconvex flux function. For this reason, these systems represent some of the few examples for which the Cauchy problem has been solved for arbitrary data of bounded variation. Also, hyperbolic degeneracies appear in each of these systems. In the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves. This phenomenon turns out to be a special case of the general phenomenon of a submanifold of R" being invariant for weak solutions of a given system of n conservation laws. Invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations; i.e., when the concentrations C, of any of the species vanish, the system must reduce to a lower order system that expresses the conservation of the remaining species. Thus C, = 0 defines a manifold which is invariant in the sense that solutions that start on the manifold, remain there for all time. Moreover, such an invariant manifold of dimension n — 1 is the boundary of an «-dimensional invariant region in the sense of Chueh, Conley and Smoller (14). In the case of

233 citations


Journal ArticleDOI
TL;DR: In this article, an experimentally verified hydrodynamic model was developed to predict solids circulation around a jet in a fluidized bed gasifier, and the growth, propagation, and collapse of bubbles were calculated.
Abstract: The object of this investigation is to develop an experimentally verified hydrodynamic model to predict solids circulation around a jet in a fluidized bed gasifier. Hydrodynamic models of fluidization use the principles of conservation of mass, momentum, and energy. To account for unequal velocities of solid and fluid phases, separate phase momentum balances are developed. Other fluid bed models used in the scale-up of gasifiers do not employ the principles of conservation of momentum. Therefore, these models cannot predict fluid and particle motion. In such models solids mixing is described by means of empirical transfer coefficients. A two-dimensional unsteady-state computer code was developed to give gas and solid velocities, void fractions, and pressure in a fluid bed with a jet. The growth, propagation, and collapse of bubbles was calculated. Time averaged voi fractions were calculated that showed good agreement with void fractions measured with a ..gamma..-ray densitometer. Calculated gas and solid velocities in the jet appeared to be reasonable.

199 citations


Journal ArticleDOI
C. M. Crowe1
TL;DR: In this paper, a projection matrix is constructed which can be used to decompose the linear problem into the solution of two subproblems, by first removing each balance around process units with an unmeasured component flow rate.
Abstract: Flow rate measurements in a steady-state process are reconciled by weighted least squares so that the conservation laws are obeyed. A projection matrix is constructed which can be used to decompose the linear problem into the solution of two subproblems, by first removing each balance around process units with an unmeasured component flow rate. The remaining measured flow rates are reconciled, and the unmeasured flow rates can then be obtained from the solution of the conservation equations. The basic case contains constraints which are linear in the component and the total flow rates. The method is extended to cases with bilinear constraints, involving unknown parameters such as split fractions. Chi-square and normal statistics are used to test for overall gross measurement errors, for gross error in each node imbalance which is fully measured, and for each measurement adjustment.

193 citations


Journal ArticleDOI
TL;DR: An explicit time differencing technique is introduced to approximate nonlinear conservation laws and convergence to the correct physical solution is proven given only a local CFL condition.
Abstract: Numerical approximations to the initial value problem for nonlinear systems of conservation laws are considered The considered system is said to be hyperbolic when all eigenvalues of every real linear combination of the Jacobian matrices are real Solutions may develop discontinuities in finite time, even when the initial data are smooth In the investigation, explicit finite difference methods which use locally varying time grids are considered The global CFL restriction is replaced by a local restriction The numerical flux function is studied from a finite volume viewpoint, and a differencing technique is developed at interface points between regions of distinct time increments

Journal ArticleDOI
TL;DR: The Hamiltonian principle of mechanics has special advantages as the beginning point for approximations as mentioned in this paper, and it easily accommodates moving disconnecting fluid boundaries and preserves the corresponding conservation laws.
Abstract: Hamilton's principle of mechanics has special advantages as the beginning point for approximations. First, it is extremely succinct. Secondly, it easily accommodates moving disconnecting fluid boundaries. Thirdly, approximations – however strong – that maintain the symmetries of the Hamiltonian will automatically preserve the corresponding conservation laws. For example, Hamilton's principle allows useful analytical and numerical approximations to the equations governing the motion of a homogeneous rotating fluid with free boundaries.

Proceedings ArticleDOI
01 Jan 1983
TL;DR: Numerical experiments show that this new implicit unconditionallystable high-resolution TVD scheme not only has a fairly rapid convergence rate, but also generates a highly resolved approximation to the steady-state solution.
Abstract: The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme. Previously announced in STAR as N83-23085

Journal ArticleDOI
TL;DR: In this paper, it is shown that the functional form of the scalar dissipation equation is more restrictive than has generally been assumed, and that the consistently modeled equation leads to qualitatively incorrect results for the decay of scalar fluctuations.
Abstract: Conserved, passive scalars with equal diffusivities evolve independently according to the same linear transport equation. This observation leads to independence and linearity principles that impose constraints on the construction of consistent turbulence closure approximations. It is shown that the functional form of the scalar dissipation equation is more restrictive than has generally been assumed, and that the consistently modeled equation leads to qualitatively incorrect results for the decay of scalar fluctuations. It is also shown that the scalar‐pressure‐gradient correlation can be obtained from the velocity‐pressure‐gradient correlation, and thus an independent model for the scalar correlation is not required. (This last result depends upon an additional assumption that is certainly valid for nearly homogeneous flows.)

Journal ArticleDOI
TL;DR: In this article, the authors developed the complete hydrodynamics of rotating superfluid4He, and other superfluids with scalar order parameters, taking into account the elasticity effects of the vortex lattice.
Abstract: This is the first in a series of papers in which we develop the complete hydrodynamics of rotating superfluid4He, and other superfluids with scalar order parameters, taking into account the elasticity effects of the vortex lattice. The theory is capable of describing the long-wavelength Tkachenko shear waves exhibited by the vortices, as well as all phenomena contained in the usual Bekarevich-Khalatnikov hydrodynamics. In this paper we develop the basic theory, ignoring the normal component of the fluid. The conserved energy, written in terms of macroscopically averaged superfluid and vortex-line velocities, includes an elastic energy associated with shear, compressional, and line-bending deformations of the vortex array. Equations of motion for three-dimensional flow, consistent with the conservation laws for mass, energy, and vorticity, are derived; and Tkachenko modes with line bending, as well as inertial modes associated with the small vortex effective mass, are investigated. In the second paper of this series we extend this description to finite temperatures, to include dynamics of the normal fluid and dissipative effects.

Book ChapterDOI
C. M. Dafermos1
01 Jan 1983
TL;DR: Tartar as discussed by the authors proposed a functional analytic treatment of quasilinear hyperbolic equations based on the delicate theory of compensated compactness which relies upon weak a priori estimates.
Abstract: The study of systems of quasilinear hyperbolic equations that result from the balance laws of continuum physics was initiated more than a century ago yet, despite considerable progress in recent years, most of the fundamental problems in the analytical theory remain unsolved. From the outset the student of the subject encounters obstacles such as the nonexistence of globally defined smooth solutions, as a reflection of the physical phenomenon of breaking of waves and development of shocks, and the nonuniqueness of (weak) solutions in the class of discontinuous functions. Furthermore, no strong a priori estimates have yet been discovered for systems of more than one equation so that the subject is not amenable to the functional analytic techniques that have swept through the theory of partial differential equations of other types. It is only now that a functional analytic treatment of these systems is emerging, based on the delicate theory of compensated compactness which relies upon weak a priori estimates. The reader may find an account of these interesting recent developments in the article by L. Tartar in this volume.

Journal ArticleDOI
TL;DR: In this paper, a theory to explain the spontaneous formation of Mach stems in reacting shock fronts is developed, where a linearized mechanism of instability through radiating boundary waves is analyzed, and an integro-differential scalar conservation law is derived.
Abstract: A theory to explain the experimentally observed spontaneous formation of Mach stems in reacting shock fronts is developed. A linearized mechanism of instability through radiating boundary waves is analyzed. Then through weakly nonlinear asymptotics, an integro-differential scalar conservation law is derived. The asymptotic expansion relates the breakdown of solutions for this scalar equation with the spontaneous formation of the complete triple-shock, slip-line Mach stem configuration.

Journal ArticleDOI
TL;DR: In this paper, the density of the subsonic and supersonic stream functions is determined in terms of the flux, and the need of two stream functions for three dimensional calculations is briefly discussed.
Abstract: The stream function equation, in conservation form, looks similar to the full potential equation and existing methods (e.g. artificial compressibility) can be readily applied. Rotational flows can be calculated once the vorticity (due to shocks or nonuniformity) is evaluated. There are, however, two main difficulties: First, the density is not uniquely determined in terms of the flux (there are two solutions; the subsonic and the supersonic branch with a square root singularity at the sonic point). Methods to overcome this difficulty are studied and results are presented with some remarks on inviscid separation and closed stream lines. Second, the need of two stream functions for three dimensional calculations is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, a general class of ILW type equations is constructed and a Hamiltonian structure is introduced to construct an infinite number of conservation laws, which are then used to solve the ILW problem.
Abstract: A general class of the ILW type equations is constructed. We introduce a Hamiltonian structure and construct an infinite number of conservation laws.

Journal ArticleDOI
01 Nov 1983
TL;DR: In this paper, a complete classification of all conservation laws of a given system is presented, which can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on.
Abstract: 1. Conservation laws For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity.

Journal ArticleDOI
Michael Shearer1
TL;DR: The following system of conservation laws is considered in this paper, where two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other.
Abstract: The following system of conservation laws is considered: where σ: ℝ→ℝ is a smooth function monotonically increasing except in an interval. Two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other. This contrasts with the corresponding situation for strictly hyperbolic systems (σ'( u )>0 for all u ), for which the two criteria are equivalent.

Journal ArticleDOI
TL;DR: In this article, the authors studied a class of systems of order two which are non-strictly hyperbolic but genuinely nonlinear, so that contact discontinuities cannot occur.

Journal ArticleDOI
TL;DR: In this article, simple formulae describing the generation of linearly frequency swept pulses in optical fibers and their subsequent compression are derived and the appropriate initial value problem is approximatively solved using partially the inverse scattering method and conservation laws for the nonlinear Schrodinger equation.

Journal ArticleDOI
TL;DR: In this paper, a quadratic conservation law for small-amplitude quasi-geostrophic disturbances on a wavy basic state is derived for describing the three-dimensional propagation of disturbances on time-averaged flows.
Abstract: A quadratic conservation law is derived for small-amplitude quasi-geostrophic disturbances on a wavy basic state. The law may be useful for describing the three-dimensional propagation of disturbances on time-averaged flows. This parallels the use of the generalized Eliassen-Palm theorem in the description of waves propagating on zonally-averaged flows.

Journal ArticleDOI
TL;DR: In this paper, the Fourier-Fourier transform was applied to a plasma model that is a generalization of the electrostatic Vlasov-Poisson system of equations.

BookDOI
01 Jan 1983
TL;DR: In this paper, the authors introduce the concept of Bifurcation theory and apply it to the problem of nonlinear elasticity in the context of optimal control theory, including the regularity problem of Extremals of Variational Integrals.
Abstract: I Expository Lectures.- Algebraic and Topological Invariants for Reaction-Diffusion Equations.- Hyperbolic Systems of Conservation Laws.- Ill-Posed Problems in Thermoelasticity Theory Lecture 1 Twinning of Thermoelastic Materials.- Lecture 2 Problems for Infinite Elastic Prisms.- Lecture 3 St.-Venantr-s Problem for Elastic Prisms.- Nonlinear Systems in Optimal Control Theory and Related Topics.- The Regularity Problem of Extremals of Variational Integrals.- Some Aspects of the Regularity Theory for Nonlinear Elliptic Systems.- Quasilinear Elliptic Systems in Diagonal Form.- Topics in Bifurcation Theory Lecture 1 A Brief Introduction to Bifurcation Theory.- Lecture 2 Unfoldings.- Lecture 3 Symmetry in Bifurcation Theory.- The Compensated Compactness Method Applied to Systems of Conservation Laws.- II Special Sessions.- a. Problems in Nonlinear Elasticity, organized by S. S. Antman (University of Maryland).- Coercivity Conditions in Nonlinear Elasticity.- Constitutive Inequalities and Dynamic Stability in the Linear Theories of Elasticity, Thermoelasticity and Viscoelasticity.- Generalized Solutions to Conservation Laws.- Stability of the Elastica.- Group Theoretic Classification of Conservation Laws in Elasticity.- b. Applications of Bifurcation Theory to Mechanics, organized by J. E. Marsden (University of California, Berkeley).- Phase Transitions Via Bifurcation from Heteroclinic Orbits.- Bifurcation under Continuous Groups of Symmetries.- Morse Decompositions and Global Continuation of Periodic Solutions for Singularly Perturbed Delay Equations.- Bifurcation and Linearization Stability in the Traction Problem.- Singular Elliptic Eigenvalue Problems for Equations and Systems.- c. Nonelliptic Problems and Phase Transitions, organized by J. M. Ball (Heriot-Watt University).- Regularization of Non Elliptic Variational Problems.- Remarks on the Relaxation of Integrals of the Calculus of Variations.- A Diffusion Equation with a Nonmonovone Constitutive Function.- An Admissibility Criterion for Fluids Exhibiting Phase Transitions.- d. Dynamical Systems and Partial Differential Equations, organized by J. K. Hale (Brown University).- Stabilization of Solutions for a System with a Continuum of Equilibria and Distinct Diffusion Coefficients.- Relation between the Trapped Rays and the Distribution of the Eigenfrequencies of the Wave Equation in the Exterior of an Obstacle.- Stabilization Properties for Nonlinear Degenerate Parabolic Equations with Cut-Off Diffusivity.- Dynamics in Parabolic Equations - An Example.

Journal ArticleDOI
TL;DR: In this article, it was shown that it is possible to impose physically reasonable conservation-law requirements which can be realized by no Dirac-equation boundary condition, and that the conservation of certian charges is determined by the short-distance physics of the problem.
Abstract: Boundary conditions on the monopole Dirac equation are, on the one hand, determined by the requirement that the Hamiltonian be Hermitean. On the other, they enforce the conservation of certian charges at the monopole core. Which charges should be conserved is determined by the short-distance physics of the problem and the connection with boundary conditions is worked out for several cases. It is found to be possible to impose physically reasonable conservation-law requirements which can be realized by no Dirac-equation boundary conditionexclamation

Journal ArticleDOI
TL;DR: In this paper, it was shown that the two-dimensional free boundary problem for incompressible irrotational water waves without surface tension has exactly eight nontrivial conservation laws.
Abstract: The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis. Introduction. The main purpose of this paper is to prove that the free boundary problem describing the motion of gravity waves over a two-dimensional irrotational, incompressible ideal fluid in the absence of surface tension (\"water waves\") has exactly eight independent conservation laws. Extensions to three-dimensional waves, with or without surface tension are indicated, but not explicitly proven. This result carries a number of implications for the interpretation of the qualitative and quantitative properties of real water waves by soliton models such as the KortewegdeVries equation, which we discuss at length in §2. The proof of such a result must incorporate a precise definition of the concept of a conservation law for a free boundary problem, which, to my knowledge, has not appeared in the literature to date. §3 elaborates on the physical and mathematical motivations for the definition proposed here, which is more general than what one might, by analogy with the corresponding concept for systems of partial differential equations, be tempted to use. The present definition of a conservation law is formulated so as to be applicable to a wide class of free boundary problems. A second result of more general applicability is an interesting characterization of conservation laws for two-dimensional free boundary problems in which the field variables consist of a single harmonic potential. In essence, the time derivative of the conserved density must equal the sum of a divergence and an analytic contribution, the latter being the unusual feature of this result; see §5. I would like to thank T. Brooke Benjamin for the vital encouragement needed to complete this work. Received by the editors November 20, 1981 and, in revised form, May 3, 1982. 1980 Mathematics Subject Classification. Primary 35R35, 35Q20, 76B15.

01 Aug 1983
TL;DR: The goal of this work is to expand one-parameter family of explicit and implicit second-order-accurate, entropy satisfying, total variation diminishing (TVD) schemes to the multidimensional Euler equations in generalized coordinate systems.
Abstract: A one-parameter family of explicit and implicit second-order-accurate, entropy satisfying, total variation diminishing (TVD) schemes was developed by Harten. These TVD schemes were the property of not generating spurious oscillations for one-dimensional nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. Application of these methods to one- and two-dimensional fluid flows containing shocks (in Cartesian coordinates) yields highly accurate nonoscillatory numerical solutions. The goal of this work is to expand these methods to the multidimensional Euler equations in generalized coordinate systems. Some numerical results of shock waves impinging on cylindrical bodies are compared with MacCormack's method.

Journal ArticleDOI
TL;DR: In this paper, it is shown that for space-times that are radially smooth of order one in the sense of Beig & Schmidt (Communs math. 87, 65 (1982)), with asymptotically electric Weyl curvature, there exists a global concept of a twistor space at spatial infinity.
Abstract: Penrose's 'quasi-local mass and angular momentum' (Penrose, Proc. R. Soc. Lond. A 381, 53 (1982)) is investigated for 2-surfaces near spatial infinity in both linearized theory on Minkowski space and full general relativity. It is shown that for space-times that are radially smooth of order one in the sense of Beig & Schmidt (Communs math. Phys. 87, 65 (1982)), with asymptotically electric Weyl curvature, there exists a global concept of a twistor space at spatial infinity. Global conservation laws for the energy-momentum and angular momentum are obtained, and the ten conserved quantities are shown to be invariant under asymptotic coordinate transformations. The relation to other definitions is discussed briefly.