Showing papers on "Conservation law published in 1993"

An introduction to partial differential equations

Abstract: Introduction* Characteristics* Classification of Characteristics * Conservation Laws and Shocks* Maximum Principles* Distributions* Function Spaces* Sobolev Spaces * Operator Theory * Linear Elliptic Equations * Nonlinear Elliptic Equations * Energy Methods for Evolution Problems * Semigroup Methods * References * Index

Material Inhomogeneities in Elasticity

Abstract: Preface -- 1 Newton's concept of physical force -- 1.1. Newton's viewpoint -- 1.2. D' Alembert's viewpoint -- 1.3. Point particles and continua 7 -- 1.4. The modern point of view: duality -- 1.5. Lagrange versus Euler -- 2 Eshelby's concept of material force -- 2.1. Ideas from solid state physics -- 2.2. Peach-Koehler force -- 2.3. Force on a singularity -- 2.4. Energy-release rate -- 2.5. Pseudomomentum -- 2.6. Relationship with phonon and photon physics -- 3 Essentials of nonlinear elasticity theory -- 3.1. Material continuum in motion -- 3.2. Elastic me sures of strains -- 3.3. Compatibility of strains -- 3.4. Balance laws (Euler-Cauchy) -- 3.5. Balance laws (Piola-Kirchhoff) -- 3.6. Constitutive equations -- 3.7. Concluding remarks -- 4 Material balance laws and inhomogeneity -- 4.1. Fully material balance laws -- 4.2. Material inhomogeneity force and pseudomomentum -- 4.3. Interpretation of pseudomomentum -- 4.4. Four formulations of the balance of linear momentum -- 4.5. Other material balance laws -- 4.6. Comments -- 5 Elasticity as a field theory -- 5.1. Elements of field theory -- 5.2. Noether's theorem -- 5.3. Variational formulation (direct-motion description) -- 5.4. Variational formulation (inverse-motion description) -- 5.5. Other material balance laws -- 5.6. Canonical Hamiltonian formulation -- 5.7. Balance of total pseudomomentum -- 5.8. Nonsimple materals: second-gradient theory -- 5.9. Complementary-energy variational principle -- 5.10. Peach-Koehler force revisited -- 5.11. Concluding remarks -- 6 Geometrical aspects of elasticity theory -- 6.1. Material uniformity and inhomogeneity -- 6.2. Eshelby stress tensor -- 6.3. Covariant material balance law of momentum -- 6.4. Continuous distributions of dislocations -- 6.5. Variational formulation using two variations -- 6.6. Second-gradient theory -- 6.7. Continuous distributions of disclinations -- 6.8. Similarity to Einstein-Cartan gravitation theory -- 7 Material inhomogeneities and brittle fracture -- 7.1. The problem of fracture -- 7.2. Generalized Reynolds and Green-Gauss theorems -- 7.3. Global material force -- 7.4. J-integral in fracture -- 7.5. Dual I-integral in fracture -- 7.6. Variational inequality: fracture propagation criterion -- 7.7. Other material balance laws and related path-independent integrals -- 7.8. Remark on the dynamical case -- 8 Material forces in electromagnetoelasticity -- 8.1. Electromagnetic elastic solids -- 8.2. Reminder of electromagnetic equations -- 8.3. Material electromagnetic fields -- 8.4. Variational principles -- 8.5. Balance of pseudomomentum and material forces -- 8.6. Fracture in electroelasticity and magnetoelasticity -- 8.7. Geometrical aspects: material uniformity -- 8.8. Electric Peach-Koehler force -- 8.9. Example of application: piezoelectric ceramics -- 9 Pseudomomentum and quasi-particles -- 9.1. Pseudomomentum of photons and phonons -- 9.2. Electromagnetic pseudomomentum -- 9.3. Conservation laws in wave theory -- 9.4. Conservation laws in soliton theory -- 9.5. Sine-Gordon systems and topological solitons -- 9.6. Boussinesq crystal equation and pseudomomentum -- 9.7. Sine-Gordon-d'Alembert systems -- 9.8. Nonlinear Schrodinger and Zakharov systems -- 10 Material forces in anelastic materials -- 10.1. Internal variables and dissipation -- 10.2. Balance of pseudomomentum -- 10.3. Global material forces -- Bibliography and references -- Index .

A fully nonlinear characteristic method for gyrokinetic simulation

Abstract: A new scheme that evolves the perturbed part of the distribution function along a set of characteristics that solves the fully nonlinear gyrokinetic equations is presented. This low‐noise nonlinear characteristic method for particle simulation is an extension of the partially linear weighting scheme, and may be considered an improvement over existing δf methods. Some of the features of this new method include the ability to keep all nonlinearities, particularly those associated with the velocity space, the use of conventional particle loading techniques, and also the retention of the conservation properties of the original gyrokinetic system in the numerically converged limit. The new method is used to study a one‐dimensional drift wave model that isolates the parallel velocity nonlinearity. A mode coupling calculation for the saturation amplitude is given, which is in good agreement with the simulation results. Finally, the method is extended to the electromagnetic gyrokinetic equations in general geometry.

Hydrodynamical limit for a Hamiltonian system with weak noise

Abstract: Starting from a general hamiltonian system with superstable pairwise potential, we construct a stochastic dynamics by adding a noise term which exchanges the momenta of nearby particles. We prolve that, in the scaling limit, the time conserved quantities, energy, momenta and density, satisfy the Euler equation of conservation laws up to a fixed timet provided that the Euler equation has a smooth solution with a given initial data up to timet. The strength of the noise term is chosen to be very small (but nonvanishing) so that it disappears in the scaling limit.

Legendre pseudospectral viscosity method for nonlinear conservation laws

Abstract: In this paper, the Legendre spectral viscosity (SV) method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied. The authors prove that by adding a small amount of SV, bounded solutions of the Legendre SV method converge to the exact scalar entropy solution. The convergence proof is based on compensated compactness arguments, and therefore applies to certain $2 \times 2$ systems. Finally, numerical experiments for scalar as well as the one-dimensional system of gas dynamics equations are presented, which confirm the convergence of the Legendre SV method. Moreover, these numerical experiments indicate that by post-processing the SV approximation, one can recover the entropy solution within spectral accuracy.

Kinks versus Shocks

Abstract: Localized phase transitions as well as shock waves can often be modeled by material discontinuities satisfying Rankine-Hugoniot (RH) jump conditions. The use of Maxwell, Gibbs-Thompson, Hertz-Knudsen, and similar (supplementary to RH) relations in the theory of dynamic phase changes suggest that the classical system of jump conditions is at least incomplete in the case of phase transitions. While the propagation of a shock wave is completely determined by the conservations laws, the boundary conditions of the problem and the condition that the entropy increases in the process, the same is not true for the propagation of phase boundaries. Additional condition must be added to the RH conditions in order to provide sufficient data for the unique determination of the transformation process. The necessity was tacitly assumed by those who attacked the calculation of the phase boundary velocity without even trying to determine this parameter from the conservation laws and boundary conditions alone.

Zero relaxation and dissipation limits for hyperbolic conservation laws

Abstract: We are interested in hyperbolic systems of conservation laws with relaxation and dissipation, particularly the zero relaxation limit. Such a limit is of interest in several physical situations, including gas flow near thermo-equilibrium, kinetic theory with small mean free path, and viscoelasticity with vanishing memory. In this article we study hyperbolic systems of two conservation laws with relaxation. For the stable case where the equilibrium speed is subcharacteristic with respect to the frozen speeds, we illustrate for a model in viscoelasticity that no oscillation develops for the nonlinear system in the zero relaxation limit. For the marginally stable case where the equilibrium speed may equal one of the frozen speeds, we show for a model in phase transitions that no oscillation arises when the dissipation is present and goes to zero more slowly than the relaxation. Our analysis includes the construction of suitable entropy pairs to derive energy estimates. We need such energy estimates not only for the compactness properties but also for the deviation from the equilibrium of the solutions for the relaxation systems. The theory of compensated compactness is then applied to study the oscillation in the zero relaxation limit. © 1993 John Wiley & Sons, Inc.

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

Abstract: This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL1-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.

A front-tracking alternative to the random choice method

Abstract: An alternative to Glimm's proof of the existence of solutions to systems of hyperbolic conservation laws is presented. The proof is based on an idea by Dafermos for the single conservation law and in some respects simplifies Glimm's original argument. The proof is based on construction of approximate solutions of which a subsequence converges. It is shown that the constructed solution satisfies Lax's entropy inequalities. The construction also gives a numerical method for solving such systems

A unified theory of available potential energy 1

Abstract: Traditional derivations of available potential energy, in a variety of contexts, involve combining some form of mass conservation together with energy conservation. This raises the questions of why such constructions are required in the first place, and whether there is some general method of deriving the available potential energy for an arbitrary fluid system. By appealing to the underlying Hamiltonian structure of geophysical fluid dynamics, it becomes clear why energy conservation is not enough, and why other conservation laws such as mass conservation need to be incorporated in order to construct an invariant, known as the pseudoenergy, that is a positive‐definite functional of disturbance quantities. The available potential energy is just the non‐kinetic part of the pseudoenergy, the construction of which follows a well defined algorithm. Two notable features of the available potential energy defined thereby are first, that it is a locally defined quantity, and second, that it is inherently...

Conservation laws for primitive equations models with inhomogeneous layers

Abstract: A general primitive equation model with non-uniform layers is set up by simply vertically averaging the density, horizontal pressure gradient and velocity fields in ech layer; these averaged fields remain a function of horizontal position and time. The evolution equations are those of the model with homogeneous layers, with the addition of a rotational horizontal force proportional to the density gradient. Potential vorticity is not conserved because of this extra term or, alternatively, because of the loss of information on the vertical shear, produced by the averaging. However, energy and momenta are conserved, as well as an infinite number of Casimirs, which depend on arbitrary functions of density, rather than potential vorticity.

Stability of travelling waves for non-convex scalar viscous conservation laws

Abstract: Travelling waves of a viscous conservation law (so-called viscous profiles) are shown to be stable in polynomialy weighted L∞ spaces. There is no assumption of convexity on the nonlinear term and thus earlier results of Il'in and Oleǐnik, Sattinger, and Kawashima and Matsumura are generalized. The method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula. Estimates are derived for the semigroup through a new technique for estimating the resolvent. © 1993 John Wiley & Sons, Inc.

Numerical methods for hyperbolic conservation laws with stiff relaxation I: spurious solutions

Abstract: The author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock-capturing finite difference scheme on a fixed, uniform spatial grid. It is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system. A second criterion is that a certain “subcharacteristic”condition be satisfied by the hyperbolic system. This conjecture is supported with analytical and numerical results for a specific example, the solution of generalized Riemann problems of a model system of equations with a fractional step scheme in which Godunov’s method is coupled with the backward Euler method. Similar ideas are applied to the numerical solution of stiff detonation problems.

On the free overfall

Abstract: The phenomenon of the free overfall, experimentally investigated since the pioneering work of H. Rouse (1933) but not enough known analytically, is studied here assuming the steady flow to be two-dimensional, irrotational and frictionless and accounting for the presence of two free curvilinear boundaries. We derive a more general equation than the known one valid for the flow over a flat bottom. By adopting this equation together with the conservation laws, we can deduce: (a) an equation for the lower profile of the nappe, (b) the value of the brink depth and (c) an equation for the free surface profile of the flow upstream from the brink depth. The results are confirmed by the already available experimental data and by new measurements of the author.

Numerical methods for hyperbolic conservation laws with stiff relaxation II: higher-order Godunov methods

Abstract: A higher-order Godunov method is presented for hyperbolic systems of conservation laws with stiff, relaxing source terms. The goal is to develop a Godunov method that produces higher-order accurate...

Geometry and dynamics in refracting systems

Abstract: The geometric and dynamic postulates for rays in inhomogeneous optical media lead succinctly to the two Hamilton equations in regions where the inhomogeneity is smooth; at a surface of discontinuity between two smooth media, they lead to two conservation laws. One of these is the Ibn Sahl (-Snell-Descartes) law of finite refraction. The transformation due to finite refraction can be in general factorized into two simpler root transformations. These conclusions apply for mechanical as well as optical systems.

Conservation laws and correlation functions in the Luttinger liquid.

Abstract: The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in dg1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.

Spectral Viscosity Approximations to Multidimensional Scalar Conservation Laws

Abstract: The authors study the spectral viscosity (SV) method in the context of multidimensional scalar conservation laws with periodic boundary conditions. They show that the spectral viscosity, which is sufficiently small to retain the formal spectral accuracy of the underlying Fourier approximation, is large enough to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Moreover, they prove that because of the presence of the spectral viscosity, the truncation error in this case becomes spectrally small, independent of whether the underlying solution is smooth or not. Consequently, the SV approximation remains uniformly bounded and converges to a measure-valued solution satisfying the entropy condition, that is, the unique entropy solution. They also show that the SV solution has a bounded total variation, provided that the total variation of the initial data is bounded, thus confirming its strong convergence to the entropy solution. They obtain an L[sup 1] convergence rate of the usual optimal order one-half. 22 refs.

A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws

Abstract: A modified triangle based adaptive difference stencil for the numerical approximation of scalar hyperbolic conservation laws in two space dimensions is constructed. The scheme satisfies the maximum principle and approximates the flux with second-order accuracy.

Exact results for the one-dimensional self-organized critical forest-fire model

Abstract: We present the analytic solution of the self-organized critical (SOC) forest-fire model in one dimension proving SOC in systems without conservation laws by analytic means. Under the condition that the system is in the steady state and very close to the critical point, we calculate the probability that a string of [ital n] neighboring sites is occupied by a given configuration of trees. The critical exponent describing the size distribution of forest clusters is exactly [tau]=2 and does not change under certain changes of the model rules. Computer simulations confirm the analytic results.

Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory

Abstract: A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

Dynamics and kinematics of reciprocal diffusions

Abstract: This paper examines the dynamics and kinematics of reciprocal diffusions. Reciprocal processes were introduced by Bernstein in 1932, and were later studied in detail by Jamison. The reciprocal diffusions are constructed here by specifying their finite joint densities in terms of the Green’s function of a general heat operator, and an end‐point density. A path integral interpretation of the heat operator Green’s function is provided, which is used to derive a stochastic form of Newton’s law, as well as a conditional distribution for the velocity of a diffusing particle given its position. These results are then employed to derive two conservation laws expressing the conservation of mass and momentum. The conservation laws do not form a closed system of equations, in general, except for two subclasses of reciprocal diffusions, the Markov and quantum diffusions.

Smooth static solutions of the spherically symmetric Vlasov-Einstein system

Abstract: We consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of globally defined, smooth, static solutions with isotropic pressure which are asymptotically flat and have a regular center, finite total mass and finite extension of the matter

Conservation laws with discontinuous flux functions

Abstract: The author studies the initial value problem for the scalar conservation law $u_t + f(u)_x = 0$ in one spatial dimension. The flow function may be discontinuous with a finite number of jump discontinuities. This paper proves existence of a weak solution, and the proof is constructive, suggesting a numerical method for the problem.

A random matrix/transition state theory for the probability distribution of state-specific unimolecular decay rates : generalization to include total angular momentum conservation and other dynamical symmetries

Abstract: A previously developed random matrix/transition state theory (RM/TST) model for the probability distribution of state‐specific unimolecular decay rates has been generalized to incorporate total angular momentum conservation and other dynamical symmetries. The model is made into a predictive theory by using a semiclassical method to determine the transmission probabilities of a nonseparable rovibrational Hamiltonian at the transition state. The overall theory gives a good description of the state‐specific rates for the D2CO→D2+CO unimolecular decay; in particular, it describes the dependence of the distribution of rates on total angular momentum J. Comparison of the experimental values with results of the RM/TST theory suggests that there is mixing among the rovibrational states.

The Shanmugadhasan canonical transformation, function groups and the extended second Noether theorem

Abstract: After the definition of a class of well-behaved singular Lagrangians, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made. The phase-space reformulation contains arbitrary first- and second-class constraints. An answer to the problem of the Dirac conjecture is given for this class of singular Lagrangians. By using the concepts of function groups and of the associated Shanmugadhasan canonical transformations, an attempt is made to arrive at a global formulation of the theorem, in which the original invariance under an “infinite continuous group” of transformations is replaced by weak quasi-invariance under an “infinite continuous group ,” whose algebra is an involutive distribution of Lie-Backlund vector fields generating the Noether transformations. Its phase-space counterpart is the involutive distribution associated with a special function group Ḡpm, which contains a function subgroup Ḡp connected (when in canonical form) to the Shanmugadhasan canonical transformations. Also, the various possible first-order formalisms are analyzed.