Showing papers on "Conservation law published in 1986"
TL;DR: In this paper, a split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrodinger equation and the space variable is discretized by means of a finite difference and a Fourier method.
Abstract: A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrodinger equation. The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.
379 citations
362 citations
TL;DR: Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions in this article, and it is shown that the symmetry properties of the coupling affect the qualitative form of the solution and the scaling behavior of the system.
Abstract: Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is “diffusive” or “I synaptic”. terms defined in the paper. The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two-point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behavior is discussed briefly.
307 citations
TL;DR: In this article, the relativistic thermodynamics of degenerate gases is presented as a field theory of the 14 fields of particle density and stress, and the field equations are based on the conservation laws of particle numbers and energy-momentum and on a balance of fluxes.
262 citations
01 Jan 1986
TL;DR: In this paper, the authors considered single-server, multi-queue systems with cyclic service and derived a pseudo-conservation law for a weighted sum of the mean waiting times at the various queues.
Abstract: This paper considers single-server, multi-queue systems with cyclic service. Non-zero switch-over times of the server between consecutive queues are assumed. A stochastic decomposition for the amount of work in such systems is obtained. This decomposition allows a short derivation of a ‘pseudo-conservation law' for a weighted sum of the mean waiting times at the various queues. Thus several recently proved conservation laws are generalised and explained.
213 citations
TL;DR: In this article, the conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method.
Abstract: The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.
182 citations
TL;DR: The local instant formulation of mass, momentum and energy conservations of two-phase flow has been developed in this paper, where the source terms at the interface are defined in terms of the local instant interfacial area concentration.
181 citations
01 Jan 1986
TL;DR: Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case, and hence convergent.
Abstract: A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.
160 citations
TL;DR: It is shown that the Hamiltonian of the 1D Hubbard model commutes with a one-parameter family of transfer matrices of a new 2D classical model corresponding to two coupled six-vertex models, a generalization of the (infinitesimal) startriangle relation.
Abstract: We show that the Hamiltonian of the 1D Hubbard model commutes with a one-parameter family of transfer matrices of a new 2D classical model corresponding to two coupled six-vertex models. Central to this result is a new local algebraic relation, a generalization of the (infinitesimal) startriangle relation.
134 citations
TL;DR: It is shown that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics.
Abstract: We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approxima- tions, is accurate to O(N-1). These numerical methods for conservation laws are the first to have proven convergence rates of greater than O(fN-1/2). 1. Introduction. It is well-known that the solution of the hyperbolic conservation law,
128 citations
TL;DR: In this paper, the mathematical structure of a model for three-phase, incompressible flow in a porous medium is examined and it is shown that, in the absence of diffusive forces, the system of conservation laws describing the flow is not necessarily hyperbolic.
Abstract: In this paper we examine the mathematical structure of a model for three-phase, incompressible flow in a porous medium. We show that, in the absence of diffusive forces, the system of conservation laws describing the flow is not necessarily hyperbolic. We present an example in which there is an elliptic region in saturation space for reasonable relative permeability data. A linearized analysis shows that in nonhyperbolic regions solutions grow exponentially. However, the nonhyperbolic region, if present, will be of limited extent which inherently limits the exponential growth. To examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method. These experiments indicate that the solutions of Riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space. In particular, when initial states are outside the elliptic region the Riemann problem solution appears to stay outside the region. Further...
TL;DR: In this article, a material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations, where the quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor.
TL;DR: In this paper, a more consilient method has been developed that formally satisfies the conservation laws more closely, allowing the mass residual to be driven to lower levels on highly irregular grids.
Abstract: A study of the computation of recirculating flows using body-fitted coordinates has been conducted with a numerical algorithm developed previously. Both the consistent treatment of the continuity equation and the effects of the grid skewness on the calculated flow field have been investigated. A more consilient method has been developed that formally satisfies the conservation laws more closely, allowing the mass residual to be driven to lower levels on highly irregular grids. The new method can also be more effective in numerically damping out disturbances in the flow field as the solution progresses. Since the computed flow fields arc found to be quite insensitive to the final level of the residuals, the residuals are not a good indicator of the level of convergence; the kinetic energy of the flow field serves as a useful alternative. It is found that the effects of the excessive local mesh skewness on the overall ac~ curacy of the calculated solution are quite tolerable. This finding demonstrates the d...
TL;DR: In this paper, a decomposition of the Bianchi identities in a Riemann-Cartan space-time with or without torsion is performed to determine those gravitational theories which have automatic conservation.
Abstract: Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The Poincare gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article.
TL;DR: In this paper, a self-consistent nonlinear reduced fluid model with finite ion-gyroradius effects is presented, which is distinctive in being correct to O((ρi/a)2) and satisfying an exact, relatively simple energy conservation law.
Abstract: Reduced fluid models have become important tools for studying the nonlinear dynamics of plasma in a large aspect‐ratio tokamak. A self‐consistent nonlinear reduced fluid model, with finite ion‐gyroradius effects is presented. The model is distinctive in being correct to O(( ρi/a)2) and in satisfying an exact, relatively simple energy conservation law.
TL;DR: In this article, the conservation integrals for some interfacial cracks are applied to get the stress intensity factors in a very simple way without solving the complicated boundary value problems, and the integrals are shown to satisfy the conservation law under certain conditions on the interfaces.
TL;DR: In this paper, Cauchy et al. etablit des taux de decroissance uniformes optimaux optimaux for des solutions du probleme de cauchy avec donnees grandes for des lois de conservation paraboliques scalaires a plusieurs dimensions d'espace.
Abstract: On etablit des taux de decroissance uniformes optimaux pour des solutions du probleme de Cauchy avec donnees grandes pour des lois de conservation paraboliques scalaires a plusieurs dimensions d'espace
TL;DR: In this article, the Riemann and Cauchy problems were solved globally for a singular system of n hyperbolic conservation laws, which arises in the study of oil reservoir simulation, has only two wave speeds, and these coincide on a surface of codimension one in state space.
TL;DR: In this paper, the Navier-Stokes equation was shown to lead to the incompressible Navier Stokes equation provided the lattice has enough symmetry and the local rules for collisions between particles obey the usual conservation laws of classical mechanics.
Abstract: A lattice gas is the representation of a gas by its restriction on the nodes of a regular lattice for discrete time steps It was recently shown by Frisch, Hasslacher and Pomeau that such very simple models lead to the incompressible Navier-Stokes equation provided the lattice has enough symmetry and the local rules for collisions between particles obey the usual conservation laws of classical mechanics We present here recent results of numerical simulations to illustrate the power of this new approach to fluid mechanics which may give new tools for numerical studies and build a bridge between cellular automata theory and complex physical problems
TL;DR: The energy momentum of any asymptotically flat vacuum solution to the Einstein equations is a well-defined, conserved, Lorentz-covariant, timelike, future-pointing vector as mentioned in this paper.
Abstract: The energy momentum of any asymptotically flat vacuum solution to the Einstein equations is a well‐defined, conserved, Lorentz‐covariant, timelike, future‐pointing vector. The only requirement is that one be given asymptotically flat initial data that satisfy very weak continuity and falloff conditions; the three‐metric must go flat faster than r−1/2. A large class of such data exists, consistent with the constraints, and the constraints play a key role in guaranteeing that the energy momentum is well behaved.
01 Jan 1986
TL;DR: In this article, the authors studied the effect of the Riemann invariants on the product of the products of the nonlinear Schrodinger Equation (NSE) with respect to the concept of Compensated Compactness.
Abstract: Convection of Microstructures by Incompressible and Slightly Compressible Flows.- Oscillations in Solutions to Nonlinear Differential Equations.- Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation.- On High-Order Accurate Interpolation for Non-Oscillatory Shock Capturing Schemes.- On the Weak Convergence of Dispersive Difference Schemes.- Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws.- On the Construction of a Modulating Multiphase Wavetrain for a Perturbed KdV Equation.- Evidence of Nonuniqueness and Oscillatory Solutions in Computational Fluid Mechanics.- Very High Order Accurate TVD Schemes.- Convergence of Approximate Solutions to Some Systems of Conservative Laws: A Conjecture on the Product of the Riemann Invariants.- Applications of the Theory of Compensated Compactness.- A General Study of a Commutation Relation given by L. Tartar.- Interrelationships among Mechanics, Numerical Analysis, Compensated Compactness, and Oscillation Theory.- The Solution of Completely Integrable Systems in the Continuum Limit of the Spectral Data.- Stability of Finite-Difference Approximations for Hyperbolic Initial-Boundary-Value Problems.- Construction of a Class of Symmetric TVD Schemes.- Information About Other Volumes in this Program.
TL;DR: In this article, it is shown that the use of high-order spline spatial interpolation removes this limit without a significant increase of the computation and complexity of the algorithm in the system that only the long-wavelength, collective phenomena are important.
TL;DR: In this article, a perturbation-theoretic scheme for dynamics of valence fluctuations in rare-earth systems with unstable 4f shells was developed for many-body systems without use of the linked-cluster theorem.
Abstract: A perturbation-theoretic scheme is developed for dynamics of valence fluctuations in rare-earth systems with unstable 4f shells. The theory is formulated in close analogy to the standard Green-function method for many-body systems but without use of the linked-cluster theorem. This formulation regards hybridization between 4f and conduction-band states as perturbation and naturally incorporates the strong on-site 4f-electron correlation. Some favorable features are: (i) the approximation scheme automatically satisfies conservation laws required for response functions; (ii) realistic 4f-shell structures with crystalline-electric-field effects can be taken into account; (iii) the theory does not have divergence difficulties over the whole temperature range. In the lowest-order self-consistent approximation, explicit formulae for dynamical susceptibilities and 4f-electron density of states are presented. At high temperatures, the theory reproduces previous results obtained by the Mori method.
TL;DR: In this paper, the role of Eshelby's energy-momentum tensors is demonstrated for a much wider class of variations than hitherto, and by a new self-contained approach.
Abstract: The analysis has to do with a Green-elastic continuum at finite strain, and with the changes in total potential energy due to notional variations of any kind of material heterogeneity. The role of Eshelby's energy-momentum tensors is demonstrated for a much wider class of variations than hitherto, and by a new self-contained approach. Stress jumps at phase boundaries and interfacial dislocations are taken into account. Within this same framework it is shown further how integral identities and conservation laws can be generated straightforwardly for homogeneous media.
TL;DR: In this article, the authors studied the regularity properties of solutions of a single conservation law and proved that if the flux function f(·) is smooth and totally nonlinear in the sense that f'(·) vanishes at isolated points only, then f′(u(·, t)) is in the class of functions of locally bounded variation for all t > 0.
TL;DR: In this paper, an infinite set of conserved charges for some Z(N) symmetric quantum spin models by constructing their Lax pairs was derived, which correspond to the Potts model, Ashkin-Teller model and the particular set of self-dual Z (N) models solved by Fateev and Zamolodchikov.
TL;DR: In this paper, the symmetries of the elliptic equation were analyzed for the case of an isothermal atmosphere in a uniform gravitational field, and a model of a model magnetostatic atmosphere was constructed in which the current density J is proportional to the cube of the magnetic potential.
Abstract: The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential. Similarity solutions of the elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained from a consideration of the invariance group of the elliptic equation. The importance of symmetries of the elliptic equation also appears in the determination of conservation laws. It turns out that the elliptic equation can be written as a variational principle, and the symmetries of the variational functional lead (via Noether's theorem) to conservation laws for the equation. As an example of the application of the similarity solutions, a model magnetostatic atmosphere is constructed in which the current density J is proportional to the cube of the magnetic potential, and falls off exponentially with distance vertical to the base, with an 'e-folding' distance equal to the gravitational scale height. The solutions show the interplay between the gravitational force, the J x B force (B, magnetic field induction) and the gas pressure gradient.
01 Jan 1986
TL;DR: In this paper, a detailed discussion of recent developments, both formal and rigorous, in the theory of weakly nonlinear geometric optics for constructing asymptotic solutions of quasi-linear hyperbolic systems in one and several space variables.
Abstract: Here we give a detailed discussion of recent developments, both formal and rigorous, in the theory of weakly nonlinear geometric optics for constructing asymptotic solutions of quasi-linear hyperbolic systems in one and several space variables. This method is the main perturbation technique used in analyzing nonlinear wave motion for hyperbolic systems. The ideas for this method originated in the late 1940’s and early 1950’s in pioneering work of Landau [8], Lighthill [10], and Whitham [18]. However, it is only in very recent work [4], [5], [7], [1] that these methods have been developed through systematic self-consistent perturbation schemes for resonant and nonresonant wave problems in one and several space dimensions. Sections II and III of this paper give a detailed discussion and description of these formal perturbation methods applied to problems in 1-D and multi-D, respectively. The reader can consult the survey in [161 which reviews the literature on weakly nonlinear hyperbolic waves before 1981 and compare this treatment with the one described in sections II and III to see the recent progress in the field in constructing such formal perturbation expansions.