Showing papers on "Conservation law published in 2000"
Book•
03 Feb 2000
TL;DR: In this paper, the authors present a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws, with a focus on balance laws with dissipative source, modeling relaxation phenomena.
Abstract: This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews
2,138 citations
TL;DR: It is proved that a scalar version of the high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions.
1,685 citations
01 Jan 2000
TL;DR: A self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves, is given in this paper.
Abstract: This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader's visual intuition with over 70 figures. A set of problems, with varying difficulty, is given at the end of each chapter. These exercises are designed to verify and expand a student's understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference for the state of the art, in the field of hyperbolic systems of conservation laws. The last chapter contains a large, up to date list of references, preceded by extensive bibliographical notes.
912 citations
TL;DR: In this article, the authors derived the covariant nonlinear dynamical equations for the gravitational and matter fields on the brane, and then linearized to find the perturbation equations on the Brane.
Abstract: In Randall-Sundrum-type brane-world cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of embedding and of the free gravitational field in the five-dimensional bulk. Instead of starting from an ansatz for the metric, we derive the covariant nonlinear dynamical equations for the gravitational and matter fields on the brane, and then linearize to find the perturbation equations on the brane. The local energy-momentum corrections are significant only at very high energies. The imprint on the brane of the nonlocal gravitational field in the bulk is more subtle, and we provide a careful decomposition of this effect into nonlocal energy density, flux and anisotropic stress. The nonlocal energy density determines the tidal acceleration in the off-brane direction, and can oppose singularity formation via the generalized Raychaudhuri equation. Unlike the nonlocal energy density and flux, the nonlocal anisotropic stress is not determined by an evolution equation on the brane, reflecting the fact that brane observers cannot in general make predictions from initial data. In particular, isotropy of the cosmic microwave background may no longer guarantee a Friedmann geometry. Adiabatic density perturbations are coupled to perturbations in the nonlocal bulk field, and in general the system is not closed on the brane. But on super- Hubble scales, density perturbations satisfy a decoupled third-order equation, and can be evaluated by brane observers. Tensor perturbations on the brane are suppressed by local bulk effects during inflation, while nonlocal effects can serve as a source or a sink. Vorticity on the brane decays as in general relativity, but nonlocal bulk effects can source the gravito-magnetic field, so that vector perturbations can be generated in the absence of vorticity.
373 citations
TL;DR: A linearized implicit scheme for computing stationary solutions of the discrete-velocity BGK and BGK-ES models is developed, which is the basis of a code which can compute high altitude hypersonic flows, in 2D plane and axisymmetric geometries.
338 citations
TL;DR: In this paper, the fundamental relation between Lie-Backlund symmetry generators and conservation laws of an arbitrary differential equation is derived without regard to a Lagrangian formulation of the differential equation.
Abstract: The fundamental relation between Lie-Backlund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.
280 citations
275 citations
TL;DR: In this article, the effect of the additional terms arising from the modelling of bed slope and breadth variation is considered and a discretisation is constructed in a manner which retains an exact balance between the flux gradients and the source terms when this is appropriate.
260 citations
TL;DR: Three numerical examples for Ogden-type material models are given to illustrate the performance of the proposed conserving schemes, showing that, relative to the standard implicit mid-point rule, the conserve schemes exhibit superior numerical stability properties without a compromise in accuracy.
257 citations
TL;DR: The conservation law structure of the continuous Galerkin method is examined and local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements are established.
247 citations
TL;DR: This work considers second-order differential systems where high-frequency oscillations are generated by a linear part, and presents a frequency expansion of the solution, and discusses two invariants of the system that determine the coefficients of the frequency expansion.
Abstract: We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to the total energy and the oscillatory harmonic energy of the original system.
For the numerical solution we study a class of symmetric methods that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals.
TL;DR: In this paper, the curvature invariance of the Randall-Sundrum (RS) single brane-world solution was analyzed in the direction of the Cauchy horizon.
Abstract: We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [L. Randall and R. Sundrum, Phys. Rev. Lett. $83,$ 4690 (1999)], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the ``electric'' part of the five-dimensional Weyl tensor, denoted by ${E}_{\ensuremath{\mu}\ensuremath{
u}},$ gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the five-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behavior in the direction of the fifth dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane world. We also obtain the asymptotic behavior of ${E}_{\ensuremath{\mu}\ensuremath{
u}}$ on the brane at spatial infinity, assuming that the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned.
TL;DR: In this paper, a high-order extension of the second-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems is presented.
Abstract: We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282].
The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.
TL;DR: This paper shows that the splitting technique for relaxation schemes can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.
Abstract: Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work [Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws [Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277]. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.
10 Oct 2000
TL;DR: A high-order extension of the recently proposed second-order, semidiscrete method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems.
Abstract: We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282].
The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.
Book•
01 Jan 2000
TL;DR: In this paper, the Neutral Action method was used to construct conservation laws for dissipative systems with damping, for which no systematic procedures existed before, and this was generalized in an important and practical way to a large class of realistic systems.
Abstract: : Considerable advances have been achieved during the period reported. Conservation laws and path-independent integrals in non-homogeneous plane elastostatics have been established. Further, conservation laws for non- homogeneous bars and beams of variable cross-section, as well as for non- homogeneous plates have been constructed. These laws should permit a more direct and simple analysis of cracks and other defects in these structural elements. A significant breakthrough came in our success of constructing conservation laws for dissipative systems. Until now, it was possible to construct conservation laws only for systems which had a Lagrangian. We have now succeeded in establishing a general procedure, which we call the Neutral Action method, which allows the construction of conservation laws (and path-independent integrals) for systems with damping, for which no systematic procedures existed before. Thus we have generalized in an important and practical way the. classical, celebrated theorem of Noether to a large class of technically more realistic systems.
TL;DR: It is proved that satisfying an appropriate D-GCL is a sufficient condition for a numerical scheme to be at least first-order time-accurate on moving meshes.
TL;DR: The algorithm discussed here uses only scalar Riemann solvers and Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow.
Abstract: Convergence is established for a scalar finite difference scheme, based on the Godunov or Engquist--Osher (EO) flux, for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. Other works in this direction have established convergence for methods employing the solution of 2 × 2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkov-type entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coefficient, it is shown that these conditions imply L1-contractiveness for piecewise C1 solutions, thus extending a well-known theorem.
TL;DR: A way to construct robust numerical schemes for the computations of numerical solutions of one- and two-dimensional hyperbolic systems of balance laws by reforming the source terms as nonconservative products and treating them directly in the definition of the numerical fluxes.
Abstract: We propose a way to construct robust numerical schemes for the computations of numerical solutions of one- and two-dimensional hyperbolic systems of balance laws. In order to reduce the computational cost, we selected the family of flux vector splitting schemes. We reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations. This is applied to a 1D shallow water system with topography and to a 2D simplified model of two-phase flows with damping effects. Numerical results and comparisons with a classical centered discretizations scheme are supplied.
TL;DR: Yee et al. as mentioned in this paper proposed a generalized energy approach based on a special splitting of the flux derivative via a convex entropy function and certain homogeneous properties for the compressible Euler equations.
TL;DR: A mesoscopic modeling and simulation technique that is very close to the technique known as dissipative particle dynamics, derived from molecular dynamics by means of a systematic coarse-graining procedure that links the forces between the dissipative particles to a hydrodynamic description of the underlying molecular dynamics particles.
Abstract: We derive a mesoscopic modeling and simulation technique that is very close to the technique known as dissipative particle dynamics. The model is derived from molecular dynamics by means of a systematic coarse-graining procedure. This procedure links the forces between the dissipative particles to a hydrodynamic description of the underlying molecular dynamics (MD) particles. In particular, the dissipative particle forces are given directly in terms of the viscosity emergent from MD, while the interparticle energy transfer is similarly given by the heat conductivity derived from MD. In linking the microscopic and mesoscopic descriptions we thus rely on the macroscopic or phenomenological description emergent from MD. Thus the rules governing this form of dissipative particle dynamics reflect the underlying molecular dynamics; in particular, all the underlying conservation laws carry over from the microscopic to the mesoscopic description. We obtain the forces experienced by the dissipative particles together with an approximate form of the associated equilibrium distribution. Whereas previously the dissipative particles were spheres of fixed size and mass, now they are defined as cells on a Voronoi lattice with variable masses and sizes. This Voronoi lattice arises naturally from the coarse-graining procedure, which may be applied iteratively and thus represents a form of renormalization-group mapping. It enables us to select any desired local scale for the mesoscopic description of a given problem. Indeed, the method may be used to deal with situations in which several different length scales are simultaneously present. We compare and contrast this particulate model with existing continuum fluid dynamics techniques, which rely on a purely macroscopic and phenomenological approach. Simulations carried out with the present scheme show good agreement with theoretical predictions for the equilibrium behavior.
TL;DR: Some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations based on the relaxation schemes by S. Jin and Z. Xin are presented.
Abstract: We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.
TL;DR: In this article, first-and high-order centred methods for conservation laws are presented and convenient TVD conditions for constructing centred TVD schemes are formulated and some useful results are proved.
Abstract: New first- and high-order centred methods for conservation laws are presented. Convenient TVD conditions for constructing centred TVD schemes are then formulated and some useful results are proved. Two families of centred TVD schemes are constructed and extended to nonlinear systems. Some numerical results are also presented.
TL;DR: This work considers consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension and provides the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes.
Abstract: We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and, in general, are not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [ Northeastern Math J., 5 (1989), pp. 395--422] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase flow equation, and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present context is to show that the (strongly degenerate) discrete diffusion term is sufficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes. Finally, we make some concluding remarks about monotone difference schemes for multidimensional equations.
TL;DR: In this article, a class of wave propagation algorithms for three-dimensional conservation laws and other hyperbolic systems is developed, which are based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second-derivative terms.
TL;DR: In this article, the Ginzburg-Landau equation for the amplitude of the pattern is coupled to an equation for large-scale mode, and the envelope of these amplitude modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, the envelope broadens.
Abstract: Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a
large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual
Ginzburg--Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show
that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is
supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these
modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile.
Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered,
including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.
TL;DR: A posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions are derived.
Abstract: In this paper we shall derive a posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.
TL;DR: It is proved that for classical many-body lattice Hamiltonians in one dimension (1D) total momentum conservation implies anomalous conductivity in the sense of the divergence of the Kubo expression for the coefficient of thermal conductivity, kappa.
Abstract: Under quite general conditions, we prove that for classical many-body lattice Hamiltonians in one dimension (1D) total momentum conservation implies anomalous conductivity in the sense of the divergence of the Kubo expression for the coefficient of thermal conductivity, {kappa} . Our results provide rigorous confirmation and explanation of many of the existing ''surprising'' numerical studies of anomalous conductivity in 1D classical lattices, including the celebrated Fermi-Pasta-Ulam problem. (c) 2000 The American Physical Society.
TL;DR: In this paper, the dynamics of unstable particles within the real-time formulation of nonequilibrium field theory initiated in a previous paper are described. And the validity conditions for the resulting quantum four-phase-space kinetic theory are discussed under the perspective to treat particles with broad damping widths.