Showing papers on "Conservation law published in 1976"
01 Sep 1976
TL;DR: In this article, an implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form, which is second-order time-accurate, noniterative, and in a spatially factored form.
Abstract: Abstract An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.
1,050 citations
TL;DR: In this article, an implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form, which is second-order time-accurate, noniterative, and in a spatially factored form.
1,036 citations
TL;DR: In this article, Rosenbluth's nonlinear, approximate tokamak equations of motion were generalized to three dimensions and conservation laws were derived and a well-known form of the energy principle was recovered from the linearized equations.
Abstract: Rosenbluth’s nonlinear, approximate tokamak equations of motion are generalized to three dimensions. The equations describe magnetohydrodynamics in the low β, incompressible, large aspect ratio limit. Conservation laws are derived and a well‐known form of the energy principle is recovered from the linearized equations. The equations are solved numerically to study kink modes in tokamaks with rectangular cross section. Fixed‐boundary kink modes, for which the plasma completely fills the conducting chamber, are considered. These modes, which are marginally stable to lowest order in circular tokamaks, become unstable with large growth rates, comparable to the growth rates of free boundary kink modes. The unstable modes are found using linearized, two‐dimensional equations. The linear results are used as initial values in the nonlinear, three‐dimensional computations. The nonlinear results show that the magnetic field is perturbed only slightly, while a large amount of plasma convection takes place carrying plasma from the center of the chamber to the walls.
879 citations
TL;DR: A survey of results for the Korteweg-deVries equation can be found in this paper, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method.
Abstract: The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...
623 citations
TL;DR: In this paper, a random choice method for solving nonlinear hyperbolic systems of conservation laws is presented, rooted in Glimm's constructive proof that such systems have solutions.
377 citations
TL;DR: In contrast to the stan-dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various properties associated with it such as Backlund transformations as discussed by the authors.
Abstract: We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the standard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various properties associated with it such as Backlund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.
260 citations
TL;DR: In this article, the theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface, and a singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account.
Abstract: The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.
238 citations
01 Jan 1976
233 citations
TL;DR: In this article, the upwind differencing method was applied to a two-dimensional model of recirculating flow in a cavity with a sliding top, and it was shown that the false diffusion associated with first order upwind difference approximations can cause the numerical solution to severely misrepresent the physical transport processes.
225 citations
TL;DR: In this article, Huygen's principle is invoked to describe the scattering of waves by an obstacle of arbitrary shape immersed in an elastic medium, and conservation laws are discussed with respect to the divergence and curl of the displacement.
Abstract: Upon invoking Huygen’s principle, matrix equations are obtained describing the scattering of waves by an obstacle of arbitrary shape immersed in an elastic medium. New relations are found connecting surface tractions with the divergence and curl of the displacement, and conservation laws are discussed. When mode conversion effects are arbitrarily suppressed by resetting appropriate matrix elements to zero, the equations reduce to a simultaneous description of acoustic and electromagnetic scattering by the obstacle at hand. Unification with acoustic/electromagnetics should provide useful guidelines in elasticity. Approximate numerical equality is shown to exist between certain of the scattering coefficients for hard and soft spheres. For penetrable spheres, explicit analytical results are found for the first time.Subject Classification: [43]20.15, [43]20.30.
198 citations
TL;DR: In this paper, it was shown that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations.
TL;DR: In this article, it was shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.
Abstract: Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain a class of conservation laws associated with linear elastodynamics. These laws represent dynamical generalizations of certain path-independent integrals in elastostatics which have been of considerable recent interest. It is shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.
TL;DR: In this article, the Lagrange multipliers are used with finite elements to achieve desirable properties in the underlying approximation for elliptic boundary value problems, in which all boundary conditions are natural.
Abstract: The purpose of this paper is to show how Lagrange multipliers can be used with finite elements to achieve a number of desirable properties in the underlying approximation. For elliptic boundary value problems, variational principles can be developed in which all boundary conditions are natural. In fluid flow problems, one can endow the approximations with physically essential conservation laws.
TL;DR: In this article, a new method of constructing solutions to the Cauchy problem for nonlinear hyperbolic systems of conservation laws in one space dimension is introduced, which is called Lax [lo] method.
TL;DR: In this article, spline interpolation with a cubic space is investigated as a way of integrating the advective equation, and the integration scheme used is second-order accurate in time, and can easily he combined with can-leapfrog approximations as a practical way of exploiting the advantages of both types of approximation for general problems.
Abstract: Upstream interpolation with a cubic space is investigated as a way of integrating the advective equation. In advection tests with a cone this is found to give much better results than realized with second-order conservative centered differencing on a double resolution mesh, and used one-third the computation time and one eighth of the memory space. The phase errors are less than those of the fourth-order Arakawa scheme at double the resolution. The integration scheme used is second-order accurate in time, and can easily he combined with can “leapfrog” approximations as a practical way of exploiting the advantages of both types of approximation for general problems. The spline interpolation representation of advection should he of use where boundary conditions are not periodic and where the exact advection of a conservation law is not as important as good phase and amplitude fidelity.
TL;DR: In this paper, the authors proposed a different approach, which was intro-intrinsic to the classical method of solution of the Riemann problem, which is based on the construction of the shock and wave curves of (1.1).
TL;DR: In this article, a program for the explicit construction of the sine-Gordon and the massive Thirring model fields is presented, which only works in the phase of the model in which the infinite set of conservation laws are valid.
TL;DR: The dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations has been shown in this paper, where the method of solution uses the conservation laws, rather than the integral equations.
Abstract: We find the dominant asymptotic behavior of the solution of the nonlinear Schrodinger equation when there is one soliton and decaying oscillations. The solution behaves like the soliton near the soliton, and like the solution found in the preceding paper (I) elsewhere. The method of solution uses the conservation laws, rather than the integral equations.
TL;DR: In this article, a system of equations governing the dynamics of a fluid under the influence of radiative forces is described, where the starting point is Thomas's form of the equation of transfer, but the final equations are good only to order n/c, where n is a typical fluid speed and c is the speed of light.
Abstract: A system of equations governing the dynamics of a fluid under the influence of radiative forces. The starting point is Thomas's form of the equation of transfer, but the final equations are good only to order ..nu../c, where ..nu.. is a typical fluid speed and c is the speed of light. Continuous absorption and Thomson scattering are treated, and they influence the energy balance differently. The radiation field is described by the first two moments of the transfer equation, and the resulting system may be thought of as a two-fluid system.
01 Jan 1976
TL;DR: In this article, the theory of Ritz is applied to the equation that Hamilton called the "Law of Varying Action." Direct analytical solutions are obtained for the transient motion of beams, both conservative and non-conservative.
Abstract: The theory of Ritz is applied to the equation that Hamilton called the 'Law of Varying Action.' Direct analytical solutions are obtained for the transient motion of beams, both conservative and nonconservative. The results achieved are compared to exact solutions obtained by the use of rigorously exact free-vibration modes in the differential equations of Lagrange and to an approximate solution obtained through the application of Gurtin's principles for linear elastodynamics. A brief discussion of Hamilton's law and Hamilton's principle is followed by examples of results for both free-free and cantilever beams with various loadings.
TL;DR: In this article, the density, spin density and pair correlation functions for the spin-half Tomonaga model are derived analytically within the framework of ordinary many body theory.
Abstract: The density, spin density and pair correlation functions for the spin-half Tomonaga model are derived analytically within the framework of ordinary many body theory. The author extends and formalises the method of Dzyaloshinskii and Larkin (Sov. Phys. JETP., vol.38, p.202 (1974)). The dynamical constraint imposed by the conservation law is expressed in terms of Wick's theorem and the correlation functions are formulated as functional integrals by means of the Stratonovich-Hubbard transformation (1957-59). The associated one-body problem is explicitly soluble and the correlation functions assume the form of Gaussian integrals which are evaluated analytically.
TL;DR: In this paper, a more precise version of the basic canonical theorem, some considerations on conservation laws and their relation, a complete treatment of the stability of the models, especially with respect to the wave amplitude, a short treatment of Lagrangian theory, a stable discrete model which might be useful for numerical experiments and an extension of the method to the case of slowly varying water depth.
Abstract: In this paper, a sequel to two others [1, 2], some extensions and improvements of this earlier work are presented. Among these are: A more precise version of the proof of the basic canonical theorem, some considerations on conservation laws and their relation, a more complete treatment of the stability of the models, especially with respect to the wave amplitude, a short treatment of the Lagrangian version of the theory, a stable discrete model which might be useful for numerical experiments and an extension of the method to the case of slowly varying water depth.
TL;DR: For general 2 × 2 genuinely nonlinear conservation laws and isentropic gas dynamics equations, not necessarily convex, this paper proved uniqueness theorems of the Cauchy problem for piecewise continuous solutions with a finite number of centered rarefaction waves in each compact set.
TL;DR: In this article, a technique for deriving a finite-difference scheme to solve initial value partial-differential equations is presented, which is based on the variational method and constrains the finite difference scheme to satisfy the conservation law(s).
TL;DR: In this paper, it is shown that Ricci collineations provide an invariant classification scheme for certain types of matter fields and the conservation laws relating to particular properties of the timelike principal-curve congruence of the given matter field and the fact that certain symmetries may be admitted.
Abstract: This paper focuses attention on the important role of Ricci collineations admitted by certain relativistic matter fields and the related conservation laws that may be admitted in the corresponding Riemannian space-times. Accordingly, one of the main objects of the present investigation is to show that Ricci collineations provide an invariant classification scheme for certain types of matter fields. These results are elaborated in terms of a number of theorems connecting particular symmetry properties with special properties of the eigenvalues and eigenvectors of the matter tensor characterizing the given matter fields. Finally, consideration is given to conservation laws relating to particular properties of the timelike principal-curve congruence of the given matter field and to the fact that certain symmetries may be admitted. Some of the results obtained are applied to the Robertson-Walker cosmological metrics. It is shown that a special class of metrics, belonging to the cosmological model, admits a particular nondegenerate Ricci collineation and the corresponding conservation law relating to matter conservation. Moreover, brief consideration is given to the appropriate specialization of these results which could be of interest at the level of ordinary relativistic hydrodynamics and plasma physics in flat space-time.
TL;DR: In this paper, an expression for the density of the force which acts on an isotropic inhomogeneous medium in an electromagnetic field is derived, based on the electrodynamics of slowly moving bodies.
Abstract: The conservation laws which follow from the field equations and their relation to the energy and momentum conservation laws are discussed. On the basis of the electrodynamics of slowly moving bodies, an expression is derived for the density of the force which acts on an isotropic inhomogeneous medium in an electromagnetic field. Attention is concentrated on elucidating the difference between the energymomentum tensors of Minkowski and Abraham. It is emphasized either of these can be used in practice to consider the exchange of energy and momentum between an emitter and a medium in which the emitter is placed. However, to analyze the processes in the medium itself, Abraham's should be used because it takes into account Abraham's volume force, which acts even on a homogeneous medium (whereas according to Minkowski no force acts at all on a transparent, homogeneous medium with density-independent permittivity in an electromagnetic field).
TL;DR: In this article, an elementary presentation of classical and relativistic collision dynamics based upon the principle of conservation of momentum is given, and the concepts of mass are implicitly defined and their basic properties are rigorously derived and discussed.
Abstract: An elementary presentation is given of classical and relativistic collision dynamics based upon the principle of conservation of momentum. The concepts of mass are shown to be implicitly defined and their basic properties are rigorously derived and discussed. Luxons and tachyons are treated on the same footing as material particles.
TL;DR: In this article, a general formalism for describing two-time fluctuations in magnetized plasma is presented, where phase functions of one-body operators are written in terms of the phase space density autocorrelation function where δ N is the fluctuation in the singular Klimontovich microdensity.
Abstract: A general formalism for describing two-time fluctuations in magnetized plasma is presented. Two-time expectations of one-body operators (phase functions) are written in terms of the phase space density autocorrelation function where δ N is the fluctuation in the singular Klimontovich microdensity. It is shown that is the first member of a set of two-time quantities which collectively obeys the linearized BBGKY cumulant hierarchy in the ( X i , t ) variables, with initial conditions successively smaller in the plasma parameter . We study in detail the case of fluctuations in thermal equilibrium, although the general formalism holds also for the non-equilibrium case. To lowest order in e P , Γ obeys the linearized Vlasov equation. From this are recovered all of Rostoker's results for fluctuations excited by Cherenkov emission and absorbed by Landau damping, as well as a constructive proof of the test particle superposition principle. To first order, Γ obeys (in the Markovian approximation) the linearized Balescu-Guernsey-Lenard equation. For frequencies and wavenumbers in the hydrodynamic regime, the velocity moments of Γ obey linearized fluid equations with classical transport coefficients (i.e. essentially those computed by Braginskii in the 3-D case). It has been found that the classical theory is in disagreement with certain computer and laboratory experiments performed in strong magnetic fields. This defect is attributed to the absence in the classical theory of contributions to the collision operator, hence transport coefficients, of fluctuations long-lived on the Vlasov scale. Analogous difficulties arise in the theory of hydrodynamics in neutral fluids. To improve the plasma theory, a renormalization of the two-time hierarchy is proposed which sums selected terms from all orders in e P and thus treats the hydrodynamic fluctuations self-consistently. The resulting theory retains appropriate fluid conservation laws, thereby avoiding erroneous results encountered in certain diffusing orbit theories, when the fluid viscosity is indiscriminantly replaced by the test particle diffusion coefficient. In order to explain the results of the computer simulations, the theory is applied in part 2 to the problem of anomalous hydrodynamic contributions to the transport coefficients.
TL;DR: For nonlinear evolution equations, a canonical transformation which keeps the Hamiltonian form invariant is investigated in this paper, and it is shown that the Backlund transformation is the canonical transformation of this type.
Abstract: For nonlinear evolution equations, a canonical transformation which keeps the Hamiltonian form invariant is investigated. It is shown that the so-called Backlund transformation is the canonical transformation of this type. Group property of the canonical transformation and relations between infinitesimal canonical transformations and conservation laws are also in· vestigated. Sine-Gordon equation, Korteweg-de Vries equation and modified Korteweg-de Vries equation are considered as examples.