Showing papers on "Conservation law published in 1997"

Lectures on Nonlinear Hyperbolic Differential Equations

Abstract: In this introductory textbook, a revised and extended version of well-known lectures by L. Hormander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.

Nonlinear Partial Differential Equations for Scientists and Engineers

Abstract: Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.

Transport and conservation laws

Abstract: We study the effect of conservation laws on the finite-temperature transport properties in one-dimensional integrable quantum many-body systems. We show that the energy current is closely related to the first conservation law in these systems and therefore the thermal transport coefficients are anomalous. Using an inequality on the time decay of current correlations we show how the existence of conserved quantities implies a finite charge stiffness (weight of the zero-frequency component of the conductivity) and so ideal conductivity at finite temperatures.

Numerical schemes for conservation laws

Abstract: Initial Value Problems for Scalar Conservation Laws in 1-D. Initial Value Problems for Scalar Conservation Laws in 2-D. Initial Value Problems for Systems in 1-D. Initial Value Problems for Systems of Conservation Laws in 2-D. Initial Boundary Value Problems for Conservation Laws. Convection-Dominated Problems. List of Figures. References. Index.

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Abstract: In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the reader can understand the algorithms and code them up for applications. Sample codes are also available from the author.

An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities

Abstract: A flux splitting scheme (AUSMDV) has been constructed with an aim at removing numerical dissipation of the Van Leer-type flux vector splittings on a contact discontinuity. The obtained scheme is also recognized as an improved advection upstream splitting method (AUSM) by Liou and Steffen. The proposed scheme has the following favorable properties: accurate and robust resolution for shock and contact (steady and moving) discontinuities; conservation of enthalpy for steady flows; algorithmic simplicity; and easy extension to general conservation laws such as that for chemically reacting flows. A simple shock fix is presented to cure the numerical shock instability associated with the ``carbuncle phenomenon'' and an entropy fix to remove an expansion shock or glitch at the sonic point. Extensive numerical experiments were conducted to validate the proposed scheme for a wide range of problems, and the results are compiled for comparison with several recent upwind methods.

Direct Construction of Conservation Laws from Field Equations

Abstract: This Letter presents an algorithm to obtain all local conservation laws for any system of field equations. The algorithm uses a formula which directly generates the conservation laws and does not depend on the system having a Lagrangian formulation, in contrast to Noether's theorem which requires a Lagrangian. Several examples are considered including dissipative systems inherently having no Lagrangian.

Dissipative particle dynamics with energy conservation

Abstract: The stochastic differential equations for a model of dissipative particle dynamics with both total energy and total momentum conservation in the particle-particle interactions are presented. The corresponding Fokker-Planck equation for the evolution of the probability distribution for the system is deduced together with the corresponding fluctuation-dissipation theorems ensuring that the ab initio chosen equilibrium probability distribution for the relevant variables is a stationary solution. When energy conservation is included, the system can sustain temperature gradients and heat flow can be modeled.

Relativistic conservation laws and integral constraints for large cosmological perturbations

Abstract: For every mapping of a perturbed spacetime onto a background and with any vector field \ensuremath{\xi} we construct a strongly, identically conserved covariant vector density $I(\ensuremath{\xi}),$ which is the divergence of a covariant antisymmetric tensor density, a ``superpotential.'' $I(\ensuremath{\xi})$ is linear in the energy-momentum tensor perturbations of matter, which may be large; $I(\ensuremath{\xi})$ does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating strongly conserved vectors over a part \ensuremath{\Sigma} of a hypersurface $S$ of the background, which spans a two-surface \ensuremath{\partial}\ensuremath{\Sigma}, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on \ensuremath{\Sigma} and, on the other, the boundary values on \ensuremath{\partial}\ensuremath{\Sigma}. We show that there are as many such integral relations as there are different mappings, \ensuremath{\xi}'s, \ensuremath{\Sigma}'s, and \ensuremath{\partial}\ensuremath{\Sigma}'s. For given boundary values on \ensuremath{\partial}\ensuremath{\Sigma}, the integral relations may be interpreted as integral constraints on local initial data including the energy-momentum perturbations. Strong conservation laws expressed in terms of Killing fields $\overline{\ensuremath{\xi}}$ of the background become ``physical'' conservation laws. In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. Traschen's ``integral constraints'' for linearized spatially localized perturbations of the energy-momentum tensor are examples of conservation laws with peculiar \ensuremath{\xi} vectors whose equations are rederived here. In Robertson-Walker spacetimes, the ``integral constraint vectors'' are the Killing vectors of a de Sitter background for a special mapping.

Pointwise convergence to shock waves for viscous conservation laws

Abstract: We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal Lp convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.

Phase Segregation Dynamics In Particle Systems with Long Range Interactions II: Interface motion

Abstract: We study properties of the solutions of a family of second order integro-differential equations, which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics. We first establish existence and uniqueness as well as some properties of the instantonic solutions. Then we concentrate on formal asymptotic (sharp interface) limits. We argue that the obtained interface evolution laws (a Stefan-like problem and the Mullins-Sekerka solidification model) coincide with the ones which can be obtained in the analogous limits from the Cahn-Hilliard equation, the fourth order PDE which is the standard macroscopic model for phase segregation with one conservation law.

Analysis and Approximation of Conservation Laws with Source Terms

Abstract: We consider a conservation law of the form (CL)I>ut + f(u)x = ax, where $a(\cdot)$ is a bounded piecewise smooth source term and f an even convex function. We first characterize the solution to the Riemann problem through a new Lax-type formula. Then we prove that for $a(\cdot)$ fixed, the semigroup associated with (CL)is an L1 contraction, and we obtain an existence theorem for weak solutions to (CL). We conclude by constructing Godunov-type difference schemes and proving that these schemes are $L^\infty$ stable and have stable steady solutions similar in structure to those of (CL). Some numerical tests are reported.

Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients

Abstract: We are interested in the initial value problem This is the familiar (multi-dimensional) linear transport equation, in conservation form. We are going to assume a very weak regularity on the coefficients Obviously the characteristics X (t, x), given by have to be defined in a generalized sense. We choose the construction of Filippov, which is classically used in the context of conservation laws. Our definition requires the uniqueness of these generalized characteristics. To illustrate this point, two examples are considered. 19 refs.

An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

Abstract: The Lighthill acoustic analogy, as embodied in the Ffowcs Williams-Hawkings (FW-H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surface integration) is also available to the FW-H method if the integration surface used in the FW-H equation is not assumed impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. This means that the FW-H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.

Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws

Abstract: This paper analyzes the non-classical shock waves which arise as limits of certain diffusive-dispersive approximations to hyperbolic conservation laws. Such shocks occur for non-convex fluxes and connect regions of different convexity. They have negative entropy dissipation for a single convex entropy function, but not all convex entropies, and do not obey the classical Oleinik entropy criterion. We derive necessary conditions for the existence of non-classical shock waves, and construct them as limits of traveling-wave solutions for several diffusive-dispersive approximations.

Numerical solution of isospectral flows

Abstract: In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L' = [B(L), L]; L(0) = L 0 , where L 0 is a d x d symmetric matrix, B(L) is a skew-symmetric matrix function of L and [B, L] is the Lie bracket operator. We show that standard Runge-Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order.

Multiresolution Schemes for the Numerical Solution of 2-D Conservation Laws I

Abstract: A generalization of Harten's multiresolution algorithms to two-dimensional (2-D) hyperbolic conservation laws is presented. Given a Cartesian grid and a discretized function on it, the method computes the local-scale components of the function by recursive diadic coarsening of the grid. Since the function's regularity can be described in terms of its scale or multiresolution analysis, the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth. Instead, in those locations, the divergence of the solution is interpolated from the next coarser grid level. First, the basic 2-D essentially nonoscillatory (ENO) scheme is presented, then the 2-D multiresolution analysis is developed, and finally the subsequent scheme is tested numerically. The computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well.

Uniqueness of Weak Solutions to Systems of Conservation Laws

Abstract: Consider a strictly hyperbolic system of conservation laws in one space dimension: Relying on the existence of the Standard Riemann Semigroup generated by , we establish the uniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumption on the variation of along space-like segments.

A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities

Abstract: Action, symplecticity, signature and complex instability are fundamental concepts in Hamiltonian dynamics which can be characterized in terms of the symplectic structure. In this paper, Hamiltonian PDEs on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two-form is assigned to each space direction and time. This leads to a new geometric formulation of the conservation of wave action for linear and nonlinear Hamiltonian PDEs, and, via Stokes's theorem, a conservation law for symplecticity. Each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode. The first invariant in this family is classical Krein signature (or energy sign, when the energy is time independent) and the other (spatial) signatures are energy flux signs, leading to a classification of instabilities that includes information about directional spatial spreading of an instability. The theory is applied to several examples: the Boussinesq equation, the water-wave equations linearized about an arbitrary Stokes's wave, rotating shallow water flow and flow past a compliant surface. Some implications for non-conservative systems are also discussed.

Reappraisal of the Kelvin Helmholtz problem. Part 1. Hamiltonian structure

Abstract: This paper and Part 2 report various new insights into the classic Kelvin–Helmholtz problem which models the instability of a plane vortex sheet and the complicated motions arising therefrom. The full nonlinear version of the hydrodynamic problem is treated, with allowance for gravity and surface tension, and the account deals in precise fashion with several inherently peculiar properties of the mathematical model. The main achievement of the paper, presented in §3, is to demonstrate that the problem admits a canonical Hamiltonian formulation, which represents a novel variational definition of a functional representing perturbations in kinetic energy. The Hamiltonian structure thus revealed is then used to account systematically for relations between symmetries and conservation laws, and none of those examined appears to have been noticed before. In §4, a generalized, non-canonical Hamiltonian structure is shown to apply when the vortex sheet becomes folded, so requiring a parametric representation, as is well known to occur in the later stages of evolution from Kelvin–Helmholtz instability. Further invariant properties are demonstrated in this context. Finally, §5, the linearized version of the problem – reviewed briefly in §2.1 – is reappraised in the light of Hamiltonian structure, and it is shown how Kelvin–Helmholtz instability can be interpreted as the coincidence of wave modes characterized respectively by positive and negative values of the Hamiltonian functional representing perturbations in total energy.