Showing papers on "Conservation law published in 1992"

On undamped heat waves in an elastic solid

Abstract: This paper is concerned with thermoelastic material behavior whose constitutive response functions possess thermal features that are more general than in the usual classical thermoelasticity. After a general development of the constitutive equations in the context of both nonlinear and linear theories, attention is focused on the latter. In particular, the one-dimensional version of the equation for the determination of temperature in the linearized theory provides an easy comparative basis of its predictive capability: In one special case where the Fourier conductivity is dominant, the temperature equation reduces to the classical Fourier law of heat conduction, which does not permit the possibility of undamped thermal waves; however,'in another special case in which the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation.

The discrete energy-momentum method: conserving algorithms for nonlinear elastodynamics

Abstract: In the absence of external loads or in the presence of symmetries (i.e., translational and rotational invariance) the nonlinear dynamics of continuum systems preserves the total linear and the total angular momentum. Furthermore, under assumption met by all classical models, the internal dissipation in the system is non-negative. The goal of this work is the systematic design of conserving algorithms that preserve exactly the conservation laws of momentum and inherit the property of positive dissipation forany step-size. In particular, within the specific context of elastodynamics, a second order accurate algorithm is presented that exhibits exact conservation of both total (linear and angular) momentum and total energy. This scheme is shown to be amenable to a completely straightforward (Galerkin) finite element implementation and ideally suited for long-term/large-scale simulations. The excellent performance of the method relative to conventional time-integrators is conclusively demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion.

Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics

Abstract: It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. For linear Hamiltonian systems, it is well known that only the Crank-Nicholson scheme exactly preserves the total energy of the system. This conservation property is typically lost in the nonlinear regime. A general class of implicit time-stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. Remarkably, the actual implementation of this class of algorithms can be effectively accomplished by means of a simple two-step solution scheme which results in essentially no added computational cost relative to standard implicit methods. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given. The good performance of the proposed methodology is demonstrated by means of three numerical examples which constitute representative model problems of nonlinear elastodynamics, nonlinear rods and nonlinear shells.

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

Abstract: We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.

Multiplication of distributions

Abstract: If Ω denotes an open subset of Rn (n = 1, 2,…), we define an algebra g (Ω) which contains the space D′(Ω) of all distributions on Ω and such that C∞(Ω) is a subalgebra of G (Ω). The elements of G (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in G(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and ƒ∈OM(R2q)—a classical Schwartz notation—for any G1,…,Gq∈G(σ), we define naturally an element ƒG1,…,Gq∈G(σ). These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.

Resonant behaviour of MHD waves on magnetic flux tubes. III : effect of equilibrium flow

Abstract: The Hollweg et al. (1990) analysis of MHD surface waves in a stationary equilibrium is extended. The conservation laws and jump conditions at Alfven and slow resonance points obtained by Sakurai et al. (1990) are generalized to include an equilibrium flow, and the assumption that the Eulerian perturbation of total pressure is constant is recovered as the special case of the conservation law for an equilibrium with straight magnetic field lines and flow along the magnetic field lines. It is shown that the conclusions formulated by Hollweg et al. are still valid for the straight cylindrical case. The effect of curvature is examined.

Statistical Mechanics, Euler’s Equation, and Jupiter’s Red Spot

Abstract: A statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid is constructed which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. The equations contain all previous approximations as special limiting cases: for example the Kraichnan energy-enstrophy theory, and the Lundgren and Pointin point vortex mean-field theories are rederived. The techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell’s theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2-d Euler flow. Finally, a phenomenological theory of the weakly driven, weakly damped Euler fluid, based on weak perturbations of the equilibrium state, is presented. A very simple two-parameter model is used to illustrate the principal ideas.

Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics

Abstract: Fluidlike models have long been used to develop qualitative understanding of the drift‐wave class of instabilities (such as the ion temperature gradient mode and various trapped‐particle modes) which are prime candidates for explaining anomalous transport in plasmas. Here, the fluid approach is improved by developing fairly realistic models of kinetic effects, such as Landau damping and gyroradius orbit averaging, which strongly affect both the linear mode properties and the resulting nonlinear turbulence. Central to this work is a simple but effective fluid model [Phys. Rev. Lett. 64, 3019 (1990)] of the collisionless phase mixing responsible for Landau damping (and inverse Landau damping). This model is based on a nonlocal damping term with a damping rate ∼ vt‖k∥‖ in the closure approximation for the nth velocity space moment of the distribution function f, resulting in an n‐pole approximation of the plasma dispersion function Z. Alternatively, this closure approximation is linearly exact (and therefore physically realizable) for a particular f0 which is close to Maxwellian. ‘‘Gyrofluid’’ equations (conservation laws for the guiding‐center density n, momentum mnu∥, and parallel and perpendicular pressures p∥ and p⊥) are derived by taking moments of the gyrokinetic equation in guiding‐center coordinates rather than particle coordinates. This naturally yields nonlinear gyroradius terms and an important gyroaveraging of the shear. The gyroradius effects in the Bessel functions are modeled with robust Pade‐like approximations. These new fluid models of phase mixing and Landau damping are being applied by others to a broad range of applications outside of drift‐wave turbulence, including strong Langmuir turbulence, laser–plasma interactions, and the α‐driven toroidicity‐induced Alfven eigenmode (TAE) instability.

Viscous limits for piecewise smooth solutions to systems of conservation laws

Abstract: In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.

Nonlinear resonance in systems of conservation laws

Abstract: The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence ...

A new conservation law constructed without using either Lagrangians or Hamiltonians

Abstract: A new conservation theorem is derived. The conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only, without using either Lagrangian or Hamiltonian structures (which may even fail to exist for the equations at hand). One example and implications of the theorem on the structure of point symmetry transformations are presented.

Solution of the Cauchy problem for a conservation law with a discontinuous flux function

Abstract: The Cauchy problem is solved for a conservation law arising in oil reservoir simulation where the flux function may depend discontinuously on the space variable. To do this front tracking is used as a method of analysis.

The regularized Chapman-Enskog expansion for scalar conservation laws

Abstract: Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.

The effect of surfactants on the dynamics of phase separation

Abstract: The dynamics of phase separation in binary systems containing surfactants is investigated by means of a time-dependent Ginzburg-Landau model. The Langevin equations for two scalar fields are solved numerically in two spatial dimensions. One of these fields is the order parameter representing the local difference in the concentrations of the two phase-separating components and the second is the local surfactant concentration. Different conservation laws of the fields, which represent different models for the dynamics of phase separation, are investigated. In all models, the average domain size for intermediate to late times is characterized by anomalously slow dynamics caused by the accumulation of surfactants at the interfaces and by the concomitant decrease in the driving force. Although all models exhibit similar slow growth, the domain structure is found to depend strongly on the nature of the conservation law. Indeed, the structure factor exhibits a peak at k>0 when the order parameter is conserved, whereas the peak is found to lie at k=0 if the order parameter is not conserved. Finally, dynamical scaling in both real- and Fourier-space correlation functions for the order parameter is found during the intermediate time regime.

Supersymmetric extensions of the nonlinear Schrödinger equation: Symmetries and coverings

Abstract: A construction is proposed for a supersymmetric generalization of the cubic Schrodinger equation, resulting in two supersymmetric systems, one of which contains a free parameter. Both systems are proven to admit an infinite set of (higher‐order) local and nonlocal symmetries and a seemingly infinite set of conservation laws, the lowest‐order terms of which are given explicitly. Moreover, the theory of coverings (equivalent to the prolongation method of Wahlquist and Estabrook) is applied to both systems. Both are seen to admit an infinite‐dimensional covering algebra, the structure of which is determined explicitly, resulting in a related supersymmetric system of differential equations, as well as an auto‐Backlund transformation for each equation. This indicates the complete integrability of both systems.

Nonlinear Wave-Activity Conservation Laws and Hamiltonian Structure for the Two-Dimensional Anelastic Equations

Abstract: Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian...

The convergence rate of approximate solutions for nonlinear scalar conservation laws

Abstract: The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.

New nonlocal symmetries with pseudopotentials

Abstract: The concept of nonlocal Lie-Backlund symmetries can be generalized by including pseudopotentials. For the KdV, HD and AKNS equations, the authors calculate generalized symmetries of such a kind. The solitary wave solutions are obtained through the transformation of trivial solutions.

Hyperbolic phenomena in a strongly degenerate parabolic equation

Abstract: We consider the equation u t =(ϕ(u) ψ (u x )) x , where ϕ>0 and where ψ is a strictly increasing function with lim s→∞ ψ=ψ ∞<∞. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u t =ψ ∞ (ϕ(u)) x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.

Physical interpretation of certain invariants for vortex filament motion under LIA

Abstract: In the context of the localized induction approximation (LIA) for the motion of a thin vortex filament in a perfect fluid, the present work deals with certain conserved quantities that emerge from the Betchov–Da Rios equations. Here, by showing that these invariants belong to a countable family of polynomial invariants for the related nonlinear Schrodinger equation (NLSE), it is demonstrated how to interpret them in terms of kinetic energy, pseudohelicity, and associated Lagrangian. It is also shown that under LIA both linear momentum and angular momentum are conserved quantities and the relation between these quantities and the whole family of polynomial invariants is discussed.