Showing papers on "Conservation law published in 1978"

An Implicit Factored Scheme for the Compressible Navier-Stokes Equations

Abstract: An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross derivatives are evaluated explicitly. However, the algorithm is unconditional ly stable and, although a three-time-lev el scheme, requires only two time levels of data storage. The algorithm is constructed in a "delta" form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensiona l shock boundary-layer interaction problem.

An exact solution for a derivative nonlinear Schrödinger equation

Abstract: A method of solution for the ’’derivative nonlinear Schrodinger equation’’ i q t =−q x x ±i (q*q 2) x is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

On wave-action and its relatives

Abstract: Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary, in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves.The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows.The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.

Uniqueness of Solutions to Hyperbolic Conservation Laws.

Abstract: : It is known that conservative systems of differential equations which result from continuum mechanics (e.g. the equations of shallow water waves, fluid dynamics, magneto-fluid dynamics and certain elasticity problems) do not have unique solutions. Thus the problem arises of proving that systems of this type have only one physically meaningful solution. This report shows that there exists at most one solution satisfying an entropy condition which generalizes the second law of thermodynamics.

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

Abstract: The nonlinear Boltzmann equation for a rarefied gas is investigated in the fluid dynamical limit to the level of compressible Euler equation locally in time, as the mean free path e tends to zero. The nonlinear hyperbolic conservation laws obtained as the limit are also the first approximation of the Chapman-Enskog expansion.

Asymptotic Behavior of Solutions of Evolution Equations

Abstract: Publisher Summary This chapter presents the methods of investigation of the asymptotic behavior of solutions of evolution equations, endowed with a dissipative mechanism, based on the study of the structure of the ω-limit set of trajectories of the evolution operator generated by the equation. The dissipative mechanism usually manifests itself by the presence of a Liapunov functional, which is constant on ω -limit sets; the central idea of the approach presented in the chapter is to use this information in conjunction with properties of ω -limit sets such as invariance and minimality. The chapter discusses two examples of wave equations with weak damping for which the scheme set up by Hale applies. In particular, this requires that the Liapunov functional be continuous on phase space. The chapter explores the case of a hyperbolic conservation law that generates a semigroup on space. It also presents a survey of various applications and extensions of these ideas that may serve as a guide to those interested in learning more about the method.

On the construction and application of implicit factored schemes for conservation laws

Abstract: Efficient, noniterative, implicit finite difference algorithms are systematically developed for nonlinear conservation laws including purely hyperbolic systems and mixed hyperbolic parabolic systems. Utilization of a rational fraction or Pade time differencing formulas, yields a direct and natural derivation of an implicit scheme in a delta form. Attention is given to advantages of the delta formation and to various properties of one- and two-dimensional algorithms.

Conservation laws for classes of nonlinear evolution equations solvable by the spectral transform

Abstract: The existence of conservation laws for novel classes of nonlinear evolution equations (with linearlyx-dependent coefficients) solvable by the spectral transform is investigated. A remarkably explicit representation is moreover obtained for the conserved quantities of the “old” classes of nonlinear evolution equations (withx-independent coefficients; including the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, etc.).

New conservation laws in general-relativistic magnetohydrodynamics

Abstract: The flow of magnetized plasma is governed by a large number of coupled equations (Maxwell's, Euler's, conservation of energy and of baryon number) so that the solution of a problem in general-relativistic magnetohydrodynamics is very complicated, even if symmetries are present. We present here a number of new conservation laws which make the solution process easier. We obtain the general criteria for a flux conservation law to exist. We apply them to obtain the relativistic versions of the conservation of magnetic flux and of Kelvin's circulation theorem for an unmagnetized fluid, as well as a new flux conservation law for a charged fluid. For stationary and axial symmetry we find conservation laws for each component of the Maxwell tensor; these are valid even if the plasma is nonperfect. For perfect plasma we find magnetic generalizations of the relativistic Bernoulli theorems for an unmagnetized fluid. We also find a new conservation law without previous analog. As an application of our results we show that extraction of rotational energy from a black hole by interaction with a magnetized plasma is not possible in the stationary state. This contradicts previous conclusions based on the approximation of geodesic flow. Finally, still for stationary and axial symmetry, we find the magnetic generalization of Kelvin's circulation theorem. With its help we reduce the problem of solving for the field of flow and for the magnetic field to the solution of two equations: baryon conservation and a Hamilton-Jacobi-type equation. A by-product of our derivations is an explicit formula for the strength of the magnetic field in terms of fluid variables.

Signature of a parallel electric field in ion and electron distributions in velocity space

Abstract: Charged particle data taken by the S3-3 satellite, reported by Mizera and Fennell (1977), are presented as contours of the velocity distribution function on a velocity-space diagram. This report focuses on the analytical technique used to interpret the particle data. Details of features exhibited by the electron and ion data in the velocity-space representations are discussed in terms of a simple electrostatic acceleration model. The observed particle populations are separated in velocity-space by recognizable demarcations calculated from the conservation laws in accordance with Liouville's theorem.

A Possible Mechanism for the Formation of Hurricane Rainbands

Abstract: In a model of hurricane rainbands as linear waves on a barotropic mean vortex, it is possible to derive two conservation laws for the perturbations: both the azimuthally integrated Reynolds torque exerted by the waves and the ratio of the azimuthally integrated radial wave energy flux to the intrinsic frequency are constant with radius for a steady wave field without dissipation or cumulus heating. The latter of these conditions can be invoked to explain the amplification of a class of waves that sustains a flux of energy directed into the vortex center and one of angular momentum directed out of it. The intrinsic phase propagation in the tangential direction is against the mean flow, but it is not fast enough to prevent the waves from being advected slowly downwind in the cyclonic sense. The Doppler shift leads to an increase in the intrinsic frequency toward the center and, in consequence of the second conservation law, to an amplification of the wave energy flux, as well as a large increase in...

Conservation laws of dynamical systems via d'alembert's principle

Abstract: In this note we study a method for finding the conserved quantities of nonconservative holonomic dynamical systems. In contrast to the classical Noetherian approach, which is based upon the variational principle of Hamilton, the starting point in this note is based on the differential principle of D'Alembert which is equally valid for conservative and nonconservative systems. In the second part of this note, an attempt is made to employ symmetry properties as a vehicle for obtaining approximate solutions of linear and non-linear dynamical systems.

Accuracy and Resolution in the Computation of Solutions of Linear and Nonlinear Equations

Abstract: Publisher Summary This chapter describes the accuracy and resolution in the computation of solutions of linear and nonlinear equations. In many problems, one is presented with piecewise initial data whose discontinuities occur along surfaces. According to the theory of hyperbolic equations, solutions with such initial data are themselves piecewise with their discontinuities occurring across characteristic surface. At points away from the discontinuities, the truncation error is small. In these regions, it is reasonable to use difference approximations of high order accuracy, except for the danger that the large truncation error at the discontinuities propagates into the smooth region. According to the theory of hyperbolic conservation laws, solutions of systems of the form are in general discontinuous. The discontinuities, called shocks, need not be present in the initial values but arise spontaneously and their speed of propagations is governed by the Rankine–Hugoniot jump relation. It is more difficult to construct accurate approximations of discontinuous solutions of nonlinear equations than of linear equations.

Soluble theory of nonlinear beam‐plasma interaction

Abstract: A soluble theory of the post‐saturation portion of a beam‐plasma interaction is developed, concentrating on explaining the results of O’Neil, Winfrey, and Malmberg. Analytic progress is made possible by applying a certain constraint procedure, characterized by the ’’rotating‐bar’’ approximation, to a Hamiltonian formulation of the problem. The procedure yields, from the original N‐particle Hamiltonian H, a new, reduced Hamiltonian H, which has only two particle‐related degrees of freedom, and which maintains the conservation laws of energy and momentum possessed by H. The equations of motion coming from H still describe the self‐consistent interaction of a mode of the plasma with the beam particles, as opposed to previous work, and, because of the great reduction in the number of degrees of freedom, explicit expressions for the nonlinear frequency shift, and growth rate, of the mode can be obtained, which are in very good agreement with the simulation results of O’Neil, Winfrey, and Malmberg.

Kinetic theory of the shear viscosity of a strongly coupled classical one-component plasma

Abstract: We present an approximation to the linearized collision operator or memory function of the exact kinetic equation obeyed by the correlation function of the phase-space density of a classical one-component plasma. This approximate collision operator generalizes the well known Balescu-Guernsey-Lenard (BGL) operator to finite wavelengths, finite frequencies, and finite coupling constants. It, moreover, satisfies the necessary symmetry relations, leads to appropriate conservation laws, and fulfills its first sum rule exactly. Next we use this operator to compute the shear viscosity $\ensuremath{\eta}$ for a series of coupling constants spanning the whole fluid phase. For weak coupling we make contact with the BGL theory, while for strong coupling we confirm, at least qualitatively, the results of Vieillefosse and Hansen, who predicted a minimum in $\ensuremath{\eta}$ as a function of temperature. We also demonstrate the important role played by the sum rules in the quantitative evaluation of a transport coefficient such as $\ensuremath{\eta}$.

Sine-Gordon equation and nonlinear σ model on a lattice

Abstract: A discrete generalization of the sine-Gordon equation on a lattice is studied. The corresponding B\"acklund transformations, soliton solutions, action principle, conservation laws, and inverse scattering equations are obtained. The connection to a discrete version of the two-dimensional nonlinear $\ensuremath{\sigma}$ model is found.

Discrete sine-Gordon equations

Abstract: Various discretizations of the sine-Gordon equation are studied. Hirota's discretization scheme is extended and two alternative discretization schemes are constructed. The associated soliton solutions, B\"acklund transformations, conservation laws, and the inverse scattering equations are obtained.

A systematic approach for correcting nonlinear instabilities

Abstract: The Lax-Wendroff (L-W) difference scheme for a single conservation law has been shown to be nonlinearly unstable near stagnation points. In this paper a simple variant of L-W is devised which has all the usual properties-conservation form, three point scheme, second order accurate on smooth solutions, but which is shown rigorously to beL 2 stable for Burger's equation and which is believed to be stable in general. This variant is constructed by adding a simple nonlinear viscosity term to the usual L-W operator. The nature of the viscosity is deduced by first stabilizing, for general conservation laws, a model differential equation derived by analyzing the truncation error for L-W, keeping only terms of order (Δt)2. The same procedure is then carried out for an analogous semi-discrete model. Finally, the full L-W difference scheme is rigorously shown to be stable provided that the C.F.L. restriction ?|u j n |?0.24 is satisfied.

Internal wave solitons

Abstract: A numerical solution of the Benjamin–Ono equation for internal waves shows soliton behavior. Two and three Lorentzian solitons pass through one another unscathed. An initial Lorentzian with larger than soliton amplitude decays into solitons, with velocities predicted by the five conservation laws.