Showing papers on "Conservation law published in 1991"

Large Scale Dynamics of Interacting Particles

Abstract: Scales.- Outline.- I Classical Particles.- 1. Dynamics.- 1.1 Newtonian Dynamics.- 1.2 Boundary Conditions.- 1.3 Dynamics of Infinitely Many Particles.- 2. States of Equilibrium and Local Equilibrium.- 2.1 Equilibrium Measures, Correlation Functions.- 2.2 The Infinite Volume Limit.- 2.3 Local Equilibrium States.- 2.4 Local Stationarity.- 2.5 The Static Continuum Limit.- 3. The Hydrodynamic Limit.- 3.1 Propagation of Local Equilibrium.- 3.2 Hydrodynamic Equations.- 3.3 The Hard Rod Fluid.- 3.4 Steady States.- 4. Low Density Limit: The Boltzmann Equation.- 4.1 Low Density (Boltzmann-Grad) Limit.- 4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories.- 4.3 Convergence of the Scaled Correlation Functions.- 4.4 The Boltzmann Hierarchy.- 4.5 Time Reversal.- 4.6 Law of Large Numbers, Local Poisson.- 4.7 The H-Function.- 4.8 Extensions.- 5. The Vlasov Equation.- 6. The Landau Equation.- 7. Time Correlations and Fluctuations.- 7.1 Fluctuation Fields.- 7.2 The Green-Kubo Formula.- 7.3 Transport for the Hard Rod Fluid.- 7.4 The Fluctuating Boltzmann Equation.- 7.5 The Fluctuating Vlasov Equation.- 8. Dynamics of a Tracer Particle.- 8.1 Brownian Particle in a Fluid.- 8.2 The Stationary Velocity Process.- 8.3 Brownian Motion (Hydrodynamic) Limit.- 8.4 Large Mass Limit.- 8.5 Weak Coupling Limit.- 8.6 Low Density Limit.- 8.7 Mean Field Limit.- 8.8 External Forces and the Einstein Relation.- 8.9 Self-Diffusion.- 8.10 Corrections to Markovian Limits.- 9. The Role of Probability, Irreversibility.- II Stochastic Lattice Gases.- 1. Lattice Gases with Hard Core Exclusion.- 1.1 Dynamics.- 1.2 Stochastic Reversibility.- 1.3 Invariant Measures, Ergodicity, Domains of Attraction.- 1.4 Driven Lattice Gases.- 1.5 Standard Models.- 2. Equilibrium Fluctuations.- 2.1 Density Correlations and Bulk Diffusion.- 2.2 The Green-Kubo Formula.- 2.3 Currents.- 2.4 The Gradient Condition.- 2.5 Linear Response, Conductivity.- 2.6 Steady State Transport.- 2.7 State of Minimal Entropy Production.- 2.8 Bounds on the Conductivity.- 2.9 The Field of Density Fluctuations.- 2.10 Scaling Limit for the Density Fluctuation Field (Proof).- 2.11 Critical Dynamics.- 3. Nonequilibrium Dynamics for Reversible Lattice Gases.- 3.1 The Nonlinear Diffusion Equation.- 3.2 Hydrodynamic Limit (Proof).- 3.3 Low Temperatures.- 3.4 Weakly Driven Lattice Gases.- 3.5 Nonequilibrium Fluctuations.- 3.6 Local Equilibrium States and Minimal Entropy Production.- 3.7 Large Deviations.- 4. Nonequilibrium Dynamics of Driven Lattice Gases.- 4.1 Hyperbolic Equation of Conservation Type.- 4.2 Asymmetric Exclusion Dynamics.- 4.3 Fluctuation Theory.- 5. Beyond the Hydrodynamic Time Scale.- 5.1 Navier-Stokes Correction for Driven Lattice Gases.- 5.2 Local Structure of a Shock.- 5.2.1 Macroscopic Equation with Fluctuations.- 5.2.2 Shock in a Random Frame of Reference.- 5.2.3 Shock in Higher Dimensions.- 6. Tracer Dynamics.- 6.1 Two Component Systems.- 6.2 Tracer Diffusion.- 6.3 Convergence to Brownian Motion.- 6.4 Nearest Neighbor Jumps in One Dimension: The Case of Vanishing Self-Diffusion.- 7. Stochastic Models with a Single Conservation Law Other than Lattice Gases.- 7.1 Lattice Gases Without Hard Core/Zero Range Dynamics.- 7.2 Interacting Brownian Particles.- 7.3 Ginzburg-Landau Dynamics.- 8. Non-Hydrodynamic Limit Dynamics.- 8.1 Kinetic Limit.- 8.2 Mean Field Limit.- References.- List of Mathematical Symbols.

High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations

Abstract: Hamilton–Jacobi (H–J) equations are frequently encountered in applications, eg, in control theory and differential games H–J equations are closely related to hyperbolic conservation laws—in one space dimension the former is simply the integrated version of the latter Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations In this paper high-order essentially nonoscillatory (ENO) schemes for H–J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed

Upwind finite-difference calculation of traveltimes

Abstract: Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite‐difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first‐arrival‐time field. The upwind finite‐difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first‐order upwind finite‐difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of flu...

Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

Abstract: We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.

A new N=2 supersymmetric Korteweg–de Vries equation

Abstract: There are two known integrable N=2 space supersymmetric extensions of the KdV equation. Both can be written as Hamiltonian systems with a common Poisson structure which corresponds to the N=2 supersymmetric form of the second KdV structure. By analyzing the conservation laws of the one parameter family of equations that share this Hamiltonian structure, one finds that a third system is singled out by the existence of nontrivial conservation laws. It is conjectured to be integrable.

Local error estimates for discontinuous solutions of nonlinear hyperbolic equations

Abstract: Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.

A kinetic equation with kinetic entropy functions for scalar conservation laws

Abstract: We construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws. In particular we prove that it is well-posed uniformly in e — the microscopic scale. We also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions — analogous to Boltzmann's microscopicH-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, we prove by both — BV compactness arguments in the multidimensional case and by compensated compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a “continuum” limit, as it converges strongly with e↓0 to the unique entropy solution of the corresponding conservation law.

Numerical relativistic hydrodynamics: Local characteristic approach.

Abstract: We extend some recent Ishock capturing methodsR designed to solve nonlinear hyperbolic systems of conservation laws and which avoid the use of artifical viscosity for treating strong discontinuities to a relativistic hydrodynamics system of equations. Some standard shock-tube problems and radial accretion onto a Schwarzschild black hole are used to calibrate our code.

Numerical simulation of a steady-state electron shock wave in a submicrometer semiconductor device

Abstract: Appropriate numerical methods for steady-state simulations (including shock waves) when the electron flow is both subsonic and supersonic are addressed. The one-dimensional steady-state hydrodynamic equations will then be elliptic in the subsonic regions and hyperbolic/elliptic in the supersonic regions. A second upwind method is used for both elliptic and hyperbolic/elliptic regions. In the elliptic regions, the second upwind method is related to the Scharfetter-Gummell exponential fitting method. The hydrodynamic model consists of a set of nonlinear conservation laws for particle number, momentum, and energy, coupled to Poisson's equation for the electric potential. The nonlinear conservation laws are just the Euler equations of gas dynamics for a gas of charged particles in an electric field, with the addition of a heat conduction term. Thus the hydrodynamic model partial differential equations (PDEs) have hyperbolic, parabolic, and elliptic modes. The nonlinear hyperbolic modes support shock waves. The first numerical simulations of a steady-state electron shock wave in a semiconductor device are presented, using the hydrodynamic model. For the ballistic diode (which models the channel of a MOSFET), the shock wave is fully developed in Si (with 1-V bias) at 300 K for a 0.1- mu m channel and at 77 K for a 1.0- mu m channel. >

A higher-order Godunov method for modeling finite deformation in elastic-plastic solids

Abstract: In this paper we develop a first-order system of conservation laws for finite deformation in solids, describe its characteristic structure, and use this analysis to develop a second-order numerical method for problems involving finite deformation and plasticity. The equations of mass, momentum, and energy conservation in Lagrangian and Eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy, as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient. Particular attention is paid to the influence of a curl constraint on the deformation gradient, so that the characteristic speeds transform properly between the two frames of reference. Next, we consider models in rate-form for isotropic elastic-plastic materials with work-hardening, and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion. In spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric, we still find that the characteristic speeds are always real for elastic behavior, and essentially always real for plastic response. These results allow us to construct a second-order Godunov method for the computation of three-dimensional displacement in a one-dimensional material viewed in the Lagrangian frame of reference. We also describe a technique for the approximate solution of Riemann problems in order to determine numerical fluxes in this algorithm. Finally, we present numerical examples of the results of the algorithm.

Solution of the hydrodynamic device model using high-order nonoscillatory shock capturing algorithms

Abstract: Simulation results for the hydrodynamic model are presented for an n/sup +/-n-n/sup +/ diode by use of shock-capturing numerical algorithms applied to the transient model with subsequent passage to the steady state. The numerical method is first order in time, but of high spatial order in regions of smoothness. Implementation typically requires a few thousand time steps. These algorithms, termed essentially nonoscillatory, have been successfully applied in other contexts to model the flow in gas dynamics, magnetohydrodynamics, and other physical situations involving the conservation laws of fluid mechanics. The presented semiconductor simulations reveal temporal and spatial velocity overshot, as well as overshoot relative to an electric field induced by the Poisson equation. Shocks are observed in the transient simulations for certain low-temperature parameter regimes. >

A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

Abstract: A new numerical framework for solving conservation laws is being developed. It employs: (1) a nontraditional formulation of the conservation laws in which space and time are treated on the same footing, and (2) a nontraditional use of discrete variables such as numerical marching can be carried out by using a set of relations that represents both local and global flux conservation.

Nonlinear switching in multiple-core couplers.

Abstract: Power-controlled switching between any output ports of different multiple-core nonlinear directional couplers is predicted. The input-output characteristic is calculated both at the half-beat length and the beat length. A second conservation law has been found in addition to the power conservation law.

The convergence rate of approximate solutions for nonlinear scalar conservation laws. Final Report

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.

Lie Superalgebraic Approach to Super Toda Lattice and Generalized Super KdV Equations

Abstract: We propose a super Lax type equation based on a certain class of Lie superalgebra as a supersymmetric extension of generalized (modified) KdV hierarchy. We are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super KdV equations. Thefirst few of the conserved currents, the (modified) super KdV equation and the super Miura transformation are worked out explicitly in the case of twisted affine Lie superalgebraOSp(2/2)(2).

Generic scale invariance in classical nonequilibrium systems (invited)

Abstract: Unlike those in equilibrium, dissipative nonequilibrium systems are capable of generic scale invariance−correlations that decay algebraically in space and time for arbitrary parameter values. For one class of such systems, viz., those (such as fluids in a temperature gradient) subjected to external white noise, the existence of a conserved order parameter is believed to be a necessary and almost sufficient condition for generic scale invariance to occur. The evidence for this assertion and the few exceptions, i.e., noisy, conserving nonequilibrium systems with exponential decays of spatial correlations, are discussed using illustrative examples taken from magnetism wherever possible. Simple calculations of exponents characterizing the algebraic decays are shown. A second class of nonequilibrium systems, exemplified by models of sandpiles and earthquakes, which have also been argued to exhibit generic scale invariance, or ‘‘self‐organized criticality,’’ are briefly discussed. These systems are either noiseless or are subjected to strongly correlated external noise, and include among them at least one apparent magnetic realization. The conditions under which scale invariance can occur in this class, and in particular whether a conservation law is necessary, are still unclear.

On the derivatives of the collision map of relativistic particles

Abstract: Consider a pair of relativistic particles with momenta u and v which collide, emerging with new momenta u' and v'. The collision map (u,v) → (u',v') is studied for a fixed scattering direction w. The Jacobian is given explicitly, and it is shown that the average over the scattering directions of the derivatives of this map is bounded in one of the variables u, v.

Lax pairs galore

Abstract: A formula that yields an (apparently—but only apparently—nontrivial) Lax pair for any nonlinear evolution PDE in 1+1 dimensions possessing a local conservation law is presented. Several examples are exhibited.

Rotational degeneracy of hyperbolic systems of conservation laws

Abstract: The purpose of the paper is to study wave patterns of hyperbolic systems in which rotational symmetry creates a specific kind of degeneracy. In this situation hyperbolicity is necessarily non-strict, so that the elementary waves have interesting patterns. The discussion is centered around a theorem on existence and uniqueness of solutions of the Riemann problem. We give a unified presentation of examples from continuum mechanics

Convergence of a streamline diffusion finite element method for a conservation law with boundary conditions

Abstract: A higher order accurate shock-capturing streamline diffusion finite element method for general scalar conservation laws is analysed; convergence towards the unique solution is proved for several space dimensions with initial and boundary conditions, using a uniqueness theorem for measure valued solutions. Furthermore, some numerical results are given.

Front tracking applied to a nonstrictly hyperbolic system of conservation laws

Abstract: The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension. The front tracking method is based on approximate solutions of Riemann problems. The method is compared with the random choice scheme and the upwind scheme.

A demonstration of consistency of an entropy balance with balance of energy

Abstract: It is shown in this note that a general balance of entropy postulated previously with only a limited motivation (based on the form of the energy equation for an inviscid fluid) is consistent with, and can be derived from, a general balance of energy. In this derivation, an early form of entropy balance does not make use of invariance conditions under superposed rigid body motions. However, with the help of the latter invariance conditions, additional results are also derived which provide some insight on the structure of the basic equations in thermomechanics.