Showing papers on "Conservation law published in 2001"

Particles and fields in fluid turbulence

Abstract: The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

Abstract: We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications

Abstract: An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

Mean Flow and Turbulence Structure of Open-Channel Flow through Non-Emergent Vegetation

Abstract: The ability of turbulence models, based on two equation closure schemes (the k-e and the k-ω formulations) to compute the mean flow and turbulence structure in open channels with rigid, nonemergent vegetation is analyzed. The procedure, developed by Raupach and Shaw (1982), for atmospheric flows over plant canopies is used to transform the 3D problem into a more tractable 1D framework by averaging the conservation laws over space and time. With this methodology, form/drag related terms arise as a consequence of the averaging procedure, and do not need to be introduced artificially in the governing equations. This approach resolves the apparent ambiguity in previously reported values of the drag-related weighting coefficients in the equations for the turbulent kinetic energy and dissipation rates. The working hypothesis for the numerical models is that the flux gradient approximation applies to spatial/temporal averaged conservation laws, so that the eddy viscosity concept can be used. Numerical results ar...

Direct construction method for conservation laws of partial differential equations. Part II: General treatment

Abstract: This paper gives a general treatment and proof of the direct conservation law method presented in Part I. In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

Strong Traces for Solutions of Multidimensional Scalar Conservation Laws

Abstract: In this paper we consider multidimensional scalar conservation laws without BV estimates defined in a subset Ω⊂ℝ+×ℝ d We show that, with a non-degeneracy hypothesis on the flux, we can define a strong notion of trace at the boundary of Ω reached by L 1 convergence

Existence of blow-up solutions in the energy space for the critical generalized KdV equation

Abstract: From these conservation laws, H appears as an energy space, so that it is a natural space in which to study the solutions. Note that p = 2 is a special case for equation (2). Indeed, from the integrability theory (see Lax [14]), we have for suitable u0 (u0 and its derivatives with fast decay at infinity) an infinite number of conservation laws. The general question is to understand the dynamics induced by such equations. Local existence in time of solutions of (2) in the energy space is now well understood; see Kato [10], Ginibre and Tsutsumi [8] for the H theory (s > 32 ), Kenig, Ponce and Vega [11] for the L theory in the case of equation (1) and sharp H theory for (2), and Bourgain [3] and [4] for the periodic case.

A steady-state capturing method for hyperbolic systems with geometrical source terms

Abstract: We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity.

Stability of Multidimensional Shocks

Abstract: This series of lectures is devoted to the study of shock waves for systems of multidimensional conservation laws. In sharp contrast with one-dimensional problems, in higher space dimensions there is no general existence theorem for solutions which allow discontinuities. Our goal is to study the existence and the stability of the simplest pattern of a single wave front ∑, separating two states u + and u -, which depend smoothly on the space-time variables x. For example, our analysis applies to perturbations of planar shocks. They are special solutions given by constant states separated by a planar front. Given a multidimensional perturbation of the initial data or a small wave impinging on the front, we study the following stability problem. Is there a local solution with the same wave pattern? Similarly, a natural problem is to investigate the multidimensional stability of one-dimensional shock fronts. However, the analysis applies to much more general situations and the main subject is the study of curved fronts.

Generalized continuity equation and modified normalization in PT-symmetric quantum mechanics

Abstract: The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

A q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-d shallow water system

Abstract: The goal of this paper is to construct a rst-order upwind scheme for solving the system of partial dierential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Berm udez and V azquez-Cend on (3, 26, 27) for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The diculty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to dene a suitable numerical scheme with global upwinding, we rst consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.

Godunov methods : theory and applications

Abstract: Oleinik's E-Condition from the Viewpoint of Numerics H. Aiso. On Some New Results for Residual Distribution Schemes R. Abgrall, T.J. Barth. Simulations of Relativistic Jets with Genesis M.A. Aloy, et al. Relativistic Jets from Collapsars M.A. Aloy, et al. Exact Computation in Numerical Linear Algebra: The Discrete Fourier Transform J.A.D.W. Anderson, P.K. Sweby. Comparative Study of HLL, HLLC and Hybrid Riemann Solvers in Unsteady Compressible Flows A. Bagabir, D. Drikakis. A New Reconstruction Technique for the Euler Equations of Gas Dynamics with Source Terms P. Bartsch, A. Borzi. Colella-Glaz Splitting Scheme for Thermally Perfect Gases A. Beccantini. Meshless Particle Methods: Recent Developments for Nonlinear Conservation Laws in Bounded Domain B.B. Moussa. Application of Wave-propagation Algorithm to Two-dimensional Thermoelastic Wave Propagation in Inhomogeneous Media A. Berezovski, G.A. Maugin. Unstructured Mesh Solvers for Hyperbolic PDEs with Source Terms: Error Estimates and Mesh Quality M. Berzins, L.J.K. Durbeck. Constancy Preserving, Conservative Methods for Free-surface Models L. Bonaventura, E. Gross. Hyperbolic-elliptic Splitting for the Pseudo-compressible Euler Equations A. Bonfiglioli. Godunov Solution of Shallow Water Equations on Curvilinear and Quadtree Grids A.G.L. Borthwick, et al. A High-order-accurate Reconstruction for the Computation of Compressible Flows on Cell-vertex Triangular Grids L.A. Catalano. Numerical Experiments with Multilevel Schemes for Conservation Laws G. Chiavassa, R. Donat. Volume-of-fluid Methods for Partial Differential Equations P. Colella. Some New Godunov and Related Relaxation Methods for Two-phase Flow Problems F. Coquel, et al. Development of Genuinely Multi-dimensional Upwind Residual Distribution Schemes for the System of Eight Wave Ideal Magnetohydrodynamic Equations on Uncunstructured Grids A. Csik, et al. Application of TVD High Resolution Schemes to the Viscous Shock Tube Problem V. Daru, C. Tenaud. Comparison of Numerical Solvers with Godunov Scheme for Multicomponent Turbulent Flows E. Declercq. Godunov-type Schemes for the MHD Equations A. Dedner, et al. Absorbing Boundary Conditions for Astrophysical MHD Simulations A. Dedner, et al. About Kinetic Schemes Built in Axisymmetrical and Spherical Geometries S. Dellacherie. Lagrangian Systems of Conservation Laws and Approximate Riemann Solvers B. Despres. Intermediate Shocks in 3D MHD Bow Shock Flows H. De Sterck, S. Poedts. A Second Order Godunov-type Scheme for Naval Hydrodynamics A. Di Mascio, et al. Uniformly High-order Methods for Unsteady Incompressible Flows D. Drikakis. Application of the Finite Volume Method with Osher Scheme and Split Technique for Different Types of Flow in a Channel K.S. Erduran, V. Kutija. A-priori Estimates for a Semi-Lagrangian Scheme for the Wave Equation M. Falcone, R. Ferretti. Interstellar Shock Structures in Weakly Ionised Gases S.A.G.E. Falle. The Ghost Fluid Method for Numerical Treatment of Discontinuities and Interfaces R.P. Fedkiw. A Hybrid Primitive-Conservative Upwind Scheme for the Drift Flux Model K.K. Fjelde, K.H. Karlsen. Numerical Simulations of Relativistic Wind Accretion onto Black Holes Using Godunov-type Methods J.A. Font, et al. A Second Order Accurate, Space-time Limited, BDF Scheme for the Linear Advection Equation S.A. Forth. Multidimensional Upwind Schemes: Application to Hydraulics P. Garcia-Navarro, et al.

A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms

Abstract: The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a non-conservative reformulation of the zero-order terms of the right-hand side of the equations. In this context, we decided to use the results of Dal Maso, Le Floch and Murat about non-conservative products, and the generalized Roe matrices introduced by Toumi to derive a first-order linearized well-balanced scheme in the sense of Greenberg and Le Roux. As a main feature, this approach is able to preserve the right asymptotic behavior of the original inhomogeneous system, which is not an obvious property. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations.

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Abstract: Nonlocal conservation laws of the form ut + Lu +∇ · f( u)= 0, where −L is the generator of a Levy semigroup on L 1 (R n ), are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. L ∼ (−�) α/2 ,1 <α< 2, f( s)∼ s|s| r−1 , r = 1 + (α − 1)/n. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time t tends to infinity, determined by self-similar source solutions.  2001 Editions scientifiques et medicales Elsevier

Coalescing neutron stars -A step towards physical models - III. Improved numerics and different neutron star masses and spins

Abstract: In this paper we present a compilation of results from our most advanced neutron star merger simulations. Special aspects of these models were refered to in earlier publications (Ruffert & Janka [CITE]; Janka et al. [CITE]), but a description of the employed numerical procedures and a more complete overview over a large number of computed models are given here. The three-dimensional hydrodynamic simulations were done with a code based on the Piecewise Parabolic Method (PPM), which solves the discretized conservation laws for mass, momentum, energy and, in addition, for the electron lepton number in an Eulerian frame of reference. Up to five levels of nested cartesian grids ensure higher numerical resolution (about 0.6km) around the center of mass while the evolution is followed in a large computational volume (side length between 300 and 400km). The simulations are basically Newtonian, but gravitational-wave emission and the corresponding back-reaction on the hydrodynamic flow are taken into account. The use of a physical nuclear equation of state allows us to follow the thermodynamic history of the stellar medium and to compute the energy and lepton number loss due to the emission of neutrinos. The computed models differ concerning the neutron star masses and mass ratios, the neutron star spins, the numerical resolution expressed by the cell size of the finest grid and the number of grid levels, and the calculation of the temperature from the solution of the entropy equation instead of the energy equation. The models were evaluated for the corresponding gravitational-wave and neutrino emission and the mass loss which occurs during the dynamical phase of the merging. The results can serve for comparison with smoothed particle hydrodynamics (SPH) simulations. In addition, they define a reference point for future models with a better treatment of general relativity and with improvements of the complex input physics. Our simulations show that the details of the gravitational-wave emission are still sensitive to the numerical resolution, even in our highest-quality calculations. The amount of mass which can be ejected from neutron star mergers depends strongly on the angular momentum of the system. Our results do not support the initial conditions of temperature and proton-to-nucleon ratio needed according to recent work for producing a solar r -process pattern for nuclei around and above the peak. The improved models confirm our previous conclusion that gamma-ray bursts are not powered by neutrino emission during the dynamical phase of the merging of two neutron stars.

Basic Aspects of Hyperbolic Relaxation Systems

Abstract: This article presents a systematic consideration of hyperbolic systems of first-order partial differential equations with source terms divided by a small parameter e. Starting with von Neumann’s stability analysis, we identify several basic structural condi-tions aiming at the existence of a well-behaved limit as e tends to zero. They are a relax-ation criterion and four stability conditions for initial value problems, and a generalized Kreiss condition for initial boundary value problems. We also discuss the reasonableness of these conditions from a few aspects including well-posedness of reduced problems, jus-tification of the limit for smooth solutions, and existence of relaxation boundary layers and shock profiles. Moreover, we point out that the structural conditions are admitted by many important models from mathematical physics and numerics of conservation laws.

Renormalization in self-consistent approximation schemes at finite temperature: Theory

Abstract: Within finite temperature field theory, we show that truncated non-perturbative selfconsistent Dyson resummation schemes can be renormalized with local counter terms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym’s �-derivable concept. The scheme generates both, the renormalized self-consistent equations of motion and the closed equations for the infinite set of counter terms. At the same time the corresponding 2PI-generating functional and the thermodynamical potential can be renormalized, in consistency with the equations of motion. This guarantees the standard �-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold. The proof uses the techniques of BPHZ-renormalization to cope with the explicit and the hidden overlapping vacuum divergences.

High-Field Limit for the Vlasov-Poisson-Fokker-Planck System

Abstract: This paper is concerned with the analysis of the stability of the Vlasov-PoissonFokker-Planck system with respect to the physical constants. If the scaled thermal mean free path converges to zero and the scaled thermal velocity remains constant, then a hyperbolic limit or equivalently a high-field limit equation is obtained for the mass density. The passage to the limit as well as the existence and uniqueness of solutions of the limit equation in L 1 , global or local in time, are analyzed according to the electrostatic or gravitational character of the field and to the space dimension. In the one-dimensional case a new concept of global solution is introduced. For the gravitational field this concept is shown to be equivalent to the concept of entropy solutions of hyperbolic systems of conservation laws.

Multi-symplectic fourier pseudospectral method for the nonlinear schr ¨ odinger equation

Abstract: Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space (4) Based on their theory, we investigate the multi-symplectic Fourier pseudospectral discretization of the nonlinear Schr¨ odinger equation (NLS) on real space We show that the multi-symplectic semi-discretization of the nonlinear Schr¨ odinger equation with periodic boundary conditions has N (the number of the nodes) semi-discrete multi- symplectic conservation laws The symplectic discretization in time of the semi-discretization leads to N full- discrete multi-symplectic conservation laws We also prove a result relating to the spectral differentiation matrix Numerical experiments are included to demonstrate the remarkable local conservation properties of multi-symplectic spectral discretizations

A Difference Scheme for Conservation Laws with a Discontinuous Flux: The Nonconvex Case

Abstract: In a previous work by the author, convergence was established for a simple difference scheme approximating a scalar conservation law, where the flux was concave and had a discontinuous spatially varying coefficient [J. D. Towers, SIAM J. Numer. Anal., 38 (2000), pp. 681--698]. The main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points. Additionally, spatially varying source terms are allowed. The spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing (TVD) in time.

Critical Thresholds in a Convolution Model for Nonlinear Conservation Laws

Abstract: In this work we consider a convolution model for nonlinear conservation laws. Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers' equation and yet exhibits the conditional finite time breakdown as in the damped Burgers' equation. We show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution. The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation. We show the exact blow-up rate when the slope of the initial profile is below the lower threshold. We further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Abstract: We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of ${\cal O}(\frac{1}{\Delta t})$ , allows us to pass to a limit as $\Delta t \downarrow 0$ . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme.

Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes

Abstract: We consider the Saint-Venant system for Shallow Water which is an usual model to describe the flows in rivers, coastal areas or floodings. The hyperbolic system of conservation laws is solved on unstructured meshes using a finite volume method together with a kinetic solver.We add to this system a friction term, the role of which is important when small water depths are considered. In this paper we address the treatment of the boundary conditions, the difficulty is due to the fact that in some cases (fluvial flows) the given boundary conditions are not sufficient to apply directly the scheme, we discuss here how to treat these boundary conditions using a Riemann invariant.Some numerical results illustrate the ability of the method to treat complex problems like the filling up or the draining off of a river bed.

How Does the Electromagnetic Field Couple to Gravity, in Particular to Metric, Nonmetricity, Torsion, and Curvature?

Abstract: The coupling of the electromagnetic field to gravity is an age-old problem. Presently, there is a resurgence of interest in it, mainly for two reasons: (i) Experimental investigations are under way with ever increasing precision, be it in the laboratory or by observing outer space. (ii) One desires to test out alternatives to Einstein’s gravitational theory, in particular those of a gauge-theoretical nature, like Einstein-Cartan theory or metric-afine gravity.— A clean discussion requires a reflection on the foundations of electrodynamics. If one bases electrodynamics on the conservation laws of electric charge and magnetic flux, one finds Maxwell’s equations expressed in terms of the excitation H = (D,H) and the field strength F = (E,B) without any intervention of the metric or the linear connection of spacetime. In other words, there is still no coupling to gravity. Only the constitutive law H = functional(F) mediates such a coupling. We discuss the different ways of how metric, nonmetricity, torsion, and curvature can come into play here. Along the way, we touch on non-local laws (Mashhoon), non-linear ones (Born-Infeld, Heisenberg-Euler, Plebaśki), linear ones, including the Abelian axion (Ni), and fid a method for deriving the metric from linear electrodynamics (Toupin, Schonberg). Finally, we discuss possible non-minimal coupling schemes.