Showing papers on "Conservation law published in 1984"

High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws

Abstract: The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is expl...

Compressible fluid flow and systems of conservation laws in several space variables

Abstract: Keywords: ecoulement : compressible ; chocs ; stabilite ; chocs ; mecanique des : fluides ; ecoulement ; theorie Reference Record created on 2005-11-18, modified on 2016-08-08

Does the missing mass problem signal the breakdown of Newtonian gravity

Abstract: We consider a nonrelativistic potential theory for gravity which differs from the Newtonian theory. The theory is built on the basic assumptions of the modified dynamics, which were shown earlier to reproduce dynamical properties of galaxies and galaxy aggregates without having to assume the existence of hidden mass. The theory involves a modification of the Poisson equation and can be derived from a Lagrangian. The total momentum, angular momentum, and (properly defined) energy of an isolated system are conserved. The center-of-mass acceleration of an arbitrary bound system in a constant external gravitational field is independent of any property of the system. In other words, all isolated objects fall in exactly the same way in a constant external gravitational field (the weak equivalence principle is satisfied). However, the internal dynamics of a system in a constant external field is different from that of the same system in the absence of the external field, in violation of the strong principle of equivalence. These two results are consistent with the phenomenological requirements of the modified dynamics. We sketch a toy relativistic theory which has a nonrelativistic limit satisfying the requirements of the modified dynamics.

Riemann Solvers, the Entropy Condition, and Difference

Abstract: A condition on the numerical flux for semidiscrete approximations to scalar, nonconvex conservation laws is introduced, and shown to guarantee convergence to the correct physical solution. An equal...

High Resolution Schemes and the Entropy Condition

Abstract: A systematic procedure for constructing semidiscrete, second order accurate, variation diminishing, five-point band width, approximations to scalar conservation laws, is presented. These schemes are constructed to also satisfy a single discrete entropy inequality. Thus, in the convex flux case, we prove convergence to the unique physically correct solution. For hyperbolic systems of conservation laws, we formally use this construction to extend the first author’s first order accurate scheme, and show (under some minor technical hypotheses) that limit solutions satisfy an entropy inequality. Results concerning discrete shocks, a maximum principle, and maximal order of accuracy are obtained. Numerical applications are also presented.

Local symmetries and conservation laws

Abstract: Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations Training examples are also given

Averaged Multivalued Solutions for Scalar Conservation Laws

Abstract: A time discretization is introduced for scalar conservation laws, which consists in averaging (in an appropriate sense) the generally multivalued solution given by the classical method of characteristics. Convergence toward the physical solutions satisfying the entropy condition is proved. Several numerical schemes are deduced after a full discretization, either with respect to the space variable (various known schemes are then recognized), or with respect to the phase variable (which leads to a space grid free scheme). Generalizations are considered toward systems of conservation laws and bidimensional scalar conservation laws.

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

Abstract: For a systemY of partial differential equations, the notion of a coveringŶ ∞→Y ∞ is introduced whereY ∞ is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations ofŶ ∞ which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.

An Extended Lagrangian Theory of Semi-Geostrophic Frontogenesis

Abstract: The Lagrangian conservation law form of the semi-geostrophic equations used by Hoskins and others is studied further in two and three dimensions. A solution of the inviscid equations containing discontinuities corresponding to atmospheric fronts is shown to exist for all time under fairly general conditions, and to be unique if the potential vorticity is required to be nonnegative. Computational results show that this solution agrees with high resolution solutions of the viscous semi-geostrophic equations. The solution, however, disagrees with that obtained from the two-dimensional viscous primitive equations. An important aspect of the difference is that the semi-geostrophic solutions allow the front to propagate into the interior of the fluid while the primitive equation solutions do not. This is discussed. If correct, it may indicate a tendency for a separation effect in the atmosphere where frictional effects are present.

Solitary waves on nonlinear elastic rods. I

Abstract: In continuation of an earlier study of propagation of solitary waves on nonlinear elastic rods, numerical investigations of blowup, reflection, and fission at continuous and discontinuous variation of the cross section for the rod and reflection at the end of the rod are presented. The results are compared with predictions of conservation theorems for energy and momentum.

Conservation laws in elasticity

Abstract: In this paper the basic results involved in the application of Noether's theorem relating symmetry groups and conservation laws to the variational problems of homogeneous elastostatics are outlined. General methods and conditions for the existence of variational and generalized symmetries are presented. Applications will be considered in subsequent papers in this series.

Conservation laws in elasticity - II. Linear homogeneous isotropic elastostatics

Abstract: Application a l'elasticite statique lineaire, homogene isotrope. Determination des lois de conservation tridimensionnelles et bidimensionnelles

The large-time behavior of the scalar genuinely nonlinear Lax-Friedrichs scheme

Abstract: The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = 0 where f(u) is, say, strictly convex double dot f dot a sub asterisk 0 is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to liner problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitive insight into the large-time behavior of the numerical computation.

New Variational Formulation of Maxwell-Vlasov and Guiding Center Theories Local Charge and Energy Conservation Laws.

Abstract: A new variational formulation of Maxwell-Vlasov and related theories is given in terms of a common Lagrangian density for both the \"Vlasov particles\" and the Maxwell fields. This formulation is used to derive in a consistent way, on the one hand, correct charge and current densities and, on the other, corresponding energy and energy flux densities. All of these densities generally show in addition to particle like contributions electric polarization and magnetization terms. By some limiting procedure collisionless guiding center theories with polarization drifts included are also treated. In this way local energy conservation laws are formulated for such theories, which has not been possible up to now

Two‐Dimensional Surges and Shocks in Open Channels

Abstract: The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow, although the complex geometry of most practical problems makes the use of finite elements very desirable. A variance of the Galerkin scheme for conservation laws in two-dimensional, nearly horizontal flow, which exhibits a remarkable shock-capturing ability, is presented. The parasitic waves in the vicinity of the discontinuity commonly present in the Galerkin solution are selectively dissipated and in fact the sharpness of the front is improved by the addition of the dissipation mechanism. The method is based on discontinuous weighting functions, which introduce upwind effects in the solution while maintaining central difference accuracy. No arbitrary parameters are needed because the dissipation level is selected analytically. No higher order derivatives appear in the governing equations which simplifies the construction and execution of the scheme. Results are presented for several flow situations that give rise to spontaneous formation of surge and shock waves in two space dimensions. The accuracy of computation is verified by comparison with analytical solutions and by continuous monitoring of the conservation properties of the model.

Fascinating polymeric liquids

Abstract: Fluid dynamics is an old subject. In 1687, Isaac Newton wrote a simple equation defining the viscosity of a fluid as the coefficient of proportionality between the shear stress and the velocity gradient. Newton's equation does well at describing gases and liquids made up of “light” molecules—those of molecular weight less than about 1000. By the middle of the last century the mathematical description of the flow of such “Newtonian” fluids was well established. This description is based on use of the laws of conservation of mass and momentum.

Derivation of the Hydrodynamical Equation for the Zero-Range Interaction Process

Abstract: small volume is so large compared with the distances between molecules that it contains an infinite number of them which are in equilibrium with given characteristics (density, temperature, * * .). Of course these parameters vary with x (say p(x), T(x), * * .). If we look at the system after a time t, the system will be still locally in equilibrium with other local characteristics (p(x, t), T(x, t), * * *) varying slowly in time although each particle individually moves very rapidly. We want to derive the equation of evolution of these macroscopical parameters." However, it is surprising that this derivation is almost never based on the analysis of the movement of the particles but rather on "conservation laws." The limiting procedure that we describe below has been used by various authors [2, 8] in efforts to give rigorous treatment of the program alluded to above. Consider a system of infinitely many particles in Rd (or Xd) moving according

Exact solutions of the nonlinear Boltzmann equation

Abstract: A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at υ→∞; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.

Stable viscosities and shock profiles for systems of conservation laws

Abstract: : Many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws. With small dissipative effects neglected, typically smooth solutions must develop discontinuities (shocks) in finite time. Re-incorporating dissipation helps select those discontinuities which are physically relevant. For this purpose, many different sorts of dissipation will do; in particular, the physical viscosity is typically degenerate and not convenient. In this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities. A natural class of admissible viscosity terms is determined based on a simple linearized stability criterion. In addition, they determine a class of degenerate second order viscosity terms of physical type which are admissible. These results are applied to the equations of compressible fluid dynamics, to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction. This should be of interest to others interested in general equations of state for compressible fluids, such as those investigating phase transitions.

SU(3) symmetry of the equations of unidimensional gas flow, with arbitrary entropy distribution

Abstract: We have shown in an earlier work that, assuming a particular class of equations of state, the Euler equations of one‐dimensional gas flow are invariant under an SU(3) group of transformations, and in fact admit of a Lie group of symmetry of infinite order; they, therefore, possess an infinite number of conservation laws. We show in the present work that the SU(3) symmetrical formalism still brings about tremendous simplification and analytical order in the most general case where the equation of state is arbitrary. The six characteristic equations assume a vector form and relate two conjugate, three‐dimensional vectors U and X. The SU(3) symmetry is only broken to a minor extent through the occurrence of a multiplicative factor Γ in the equations. The conservation laws take the form of the Cauchy integrability condition for the elements of a traceless second rank tensor eij and, taken all together, form an SU(3) octet; in the most general case, however, there exist four conservation laws only (five if the...

Convergence of a large time step generalization of Godunov's method for conservation laws

Abstract: A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.

Some explicit triangular finite element schemes for the Euler equations

Abstract: Several explicit schemes are presented for triangular P0 and P1finite elements. A first-order accurate upwind P0 scheme is compared to a FLIC type method. A second-order accurate Richtmyer scheme is constructed. Applications are given for the Euler system of conservation laws in the 2-dimensional case.