Showing papers on "Conservation law published in 2005"
Book•
11 Aug 2005
TL;DR: In this article, Jacobi polynomials Gauss-type integration Collocation differentiation Co discontinuous expansion bases are used to simulate incompressible flows in one-dimensional expansion bases.
Abstract: Introduction Fundamental concepts in one dimension Multi-dimensional expansion bases Multi-dimensional formulations Diffusion equation Advection and advection-diffusion Non-conforming elements Algorithms for incompressible flows Incompressible flow simulations:verification and validation Hyperbolic conservation laws Appendices Jacobi polynomials Gauss-Type integration Collocation differentiation Co discontinuous expansion bases Characteristic flux decomposition References Index
1,278 citations
TL;DR: New implicit–explicit (IMEX) Runge–Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms are considered, with high accuracy in space and several applications are presented.
Abstract: We consider new implicit---explicit (IMEX) Runge---Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge---Kutta method (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented
505 citations
Book•
01 Jan 2005
TL;DR: In this paper, the L1 theory for Scalar conservation laws is extended to include the Cauchy problem and the stability of classical solutions. But it does not consider the Riemann problem.
Abstract: Balance Laws.- Introductioin to Continuum Physics.- Hyperbolic Systems of Balance Laws.- The Cauchy Problem.- Entropy and the Stability of Classical Solutions.- The L1 Theory for Scalar Conservation Laws.- Hyperbolic Systems of Balance Laws in One-Space Dimension.- Admissible Shocks.- Admissible Wave Fans and the Riemann Problem.- Generalized Characteristics.- Genuinely Nonlinear Scalar Conservation Law.- Genuinely Nonlinear Systems of Two Conservation Laws.- The Random Choice Method.- The Front Tracking Method and Standard Riemann Semigroups.- Construction of BV Volutions by the Vanishing Viscosity Method.- Compensated Compactness.- Bibliography.- Author Index.- Subject Index.
442 citations
TL;DR: In this paper, a link-type collision operator is introduced as an alternative to a polynomial collision basis, which can be used to eliminate the second order tensor of the diffusion.
425 citations
TL;DR: This paper designs high order finite difference WENO schemes to this system with such exact conservation property (C-property) and at the same time maintaining genuine high order accuracy.
322 citations
TL;DR: It is proved existence of solutions to the Cauchy problem and it is shown that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions.
Abstract: This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions.Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.
303 citations
TL;DR: In this paper, the authors introduce the notion of General Relative Entropy Inequality for several linear PDEs, and give several types of applications of the GRIIN: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions for periodic forcing.
276 citations
TL;DR: A model for the flow of pedestrians is presented that describes features typical of this flow, such as the fall due to panic in the outflow of people through a door.
Abstract: We present a model for the o w of pedestrians that describes features typical of this o w, such as the fall due to panic in the outo w of people through a door. The mathematical techniques essentially depend on the use of nonclassical shocks in scalar conservation laws. 2000 Mathematics Subject Classic ation: 35L65, 90B20.
241 citations
TL;DR: A new and canonical way of writing the equations of gas dynamics in Lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws is proposed and a new conservative and entropy-consistent finite-volume numerical scheme is derived.
Abstract: We propose a new and canonical way of writing the equations of gas dynamics in Lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws. One part of the system is called the physical part and contains physical variables; the other part is the geometrical part. We show that the physical part is symmetrizable. We show that the weak hyperbolicity is due to shear contact discontinuities. Free divergence constraints play an important role in the system. We prove the L
2 stability of the physical part of the system. Based on this formulation, we derive a new conservative and entropy-consistent finite-volume numerical scheme. We prove the stability of the numerical scheme. Numerical results show the potential interest of this approach. Various examples (Born-Infeld, MHD, 3D lagrangian gas dynamics) can be written using the same abstract formalism.
217 citations
TL;DR: In this paper, a new definition of a δ -shock wave type solution for a class of systems of conservation laws in the one-dimensional case was introduced, and the weak asymptotics method developed by the authors was used to construct formulas describing the propagation and interaction of δ-shock waves.
214 citations
TL;DR: In this article, the voter model dynamics in random networks with an arbitrary distribution of the degree of the nodes were considered and it was shown that for the usual node-update dynamics the average magnetization is not conserved.
Abstract: We consider the voter model dynamics in random networks with an arbitrary distribution of the degree of the nodes. We find that for the usual node-update dynamics the average magnetization is not conserved, while an average magnetization weighted by the degree of the node is conserved. However, for a link-update dynamics the average magnetization is still conserved. For the particular case of a Barabasi-Albert scale-free network, the voter model dynamics leads to a partially ordered metastable state with a finite-size survival time. This characteristic time scales linearly with system size only when the updating rule respects the conservation law of the average magnetization. This scaling identifies a universal or generic property of the voter model dynamics associated with the conservation law of the magnetization.
TL;DR: A computational study of the flow behavior of a lab-scale fluidized bed using a ‘discrete particle method’ (DPM) and a multi-fluid computational fluid dynamic (CFD) model is presented.
TL;DR: In this paper, a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equation modeling nematic liquid crystals is constructed, and a new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.
Abstract: We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equation modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.
TL;DR: In this article, the bi-Hamiltonian structure of the Camassa-Holm equation is used to show that conservation laws H-n[m] are homogeneous with respect to the scaling m --> lambdam.
Abstract: We use the bi-Hamiltonian structure of the Camassa-Holm equation to show that its conservation laws H-n[m] are homogeneous with respect to the scaling m --> lambdam. Moreover, a direct argument is presented proving that H-1, H-2,..., are of local character. Finally, simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the H(n)s.
TL;DR: For two classes of hyperbolic systems of conservation laws new definitions of a δ-shock wave type solution are introduced in this article, which give natural generalizations of the classical definition of weak solutions.
Abstract: For two classes of hyperbolic systems of conservation laws new definitions of a δ-shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of δ-shocks. The weak asymptotics method developed by the authors is used to describe the propagation of δ-shock waves to the three types of systems of conservation laws and derive the corresponding Rankine-Hugoniot conditions for 0-shocks.
TL;DR: In this article, the authors study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones.
Abstract: We study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Muller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a height-coordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi-hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure-coordinate framework are discussed. © Crown copyright 2005.
01 Jan 2005
TL;DR: An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context and the new methodology is applied to scalar transport equations discretized in space by the Galerkin method.
Abstract: An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This modification leads to an upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. Two closely related flux correction strategies are presented. The first one is based on a multidimensional generalization of total variation diminishing (TVD) schemes, whereas the second one represents an extension of the FEM-FCT paradigm to implicit time-stepping. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the M-matrix property. The dffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and inserted into the global defect vector. The new methodology is applied to scalar transport equations discretized in space by the Galerkin method. Its performance is illustrated by numerical examples for 2D benchmark problems.
TL;DR: It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution.
Abstract: A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor--Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor--Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.
TL;DR: In this article, the authors developed a general framework for discussing both isotropic-viscoplastic materials and amorphous materials based on the Kroner-Lee decomposition of the deformation gradient into elastic and inelastic parts, a system of microforces consistent with its own balance, and a mechanical version of the second law.
TL;DR: In this article, a mathematical model describing supply chains on a network is introduced, in particular conditions on each vertex of the network are specified, and this leads to a system of nonlinear conservation laws coupled to ordinary differential equations.
Abstract: A mathematical model describing supply chains on a network is introduced. In particular, conditions on each vertex of the network are specified. Finally, this leads to a system of nonlinear conservation laws coupled to ordinary differential equations. To prove the existence of a solution we make use of the front tracking method. A comparison to another approach is given and numerical results are presented.
TL;DR: In this article, the authors prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes by a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux.
Abstract: We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions
TL;DR: In this paper, a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multi-scale method (HMM) is developed, where the atomistic model is formulated in the form of conservation laws of mass, momentum and energy.
Abstract: We developa general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we developan additional strategy for defect tracking. This method naturally decoup les the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented. r 2005 Elsevier Ltd. All rights reserved.
TL;DR: The self-consistent Green's function is obtained from a conserving self-energy approximation, which implies that the observables calculated from the Green's functions agree with the macroscopic conservation laws for particle number, momentum, and energy.
Abstract: We have calculated the self-consistent Green’s function for a number of atoms and diatomic molecules. This Green’s function is obtained from a conserving self-energy approximation, which implies that the observables calculated from the Green’s functions agree with the macroscopic conservation laws for particle number, momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calculating the total energy all give the same result. In these calculations we use the finite temperature formalism and calculate the Green’s function on the imaginary time axis. This allows for a simple extension to nonequilibrium systems. We have compared the energies from self-consistent Green’s functions to those of nonselfconsistent schemes and also calculated ionization potentials from the Green’s functions by using the extended Koopmans’ theorem.
TL;DR: This paper proposes an adaptive nonlinear finite volume ADER method on unstructured triangular meshes for scalar conservation laws, which works with WENO reconstruction and is supported by numerical results concerning Burgers equation.
TL;DR: Some generalized variational principles for ion-acoustic plasma waves were obtained by He's semi-inverse method as discussed by the authors, which has profound implications in physical understandings, explaining the interaction between various variables in an energy view and the existence of conservation law.
Abstract: Some generalized variational principles are obtained for ion-acoustic plasma waves by He's semi-inverse method. The obtained variational principle has profound implications in physical understandings, explaining the interaction between various variables in an energy view and the existence of conservation law.
TL;DR: In this paper, the existence of the strong trace on the initial hyperspace t = 0 was proved for a wide class of functions including generalized entropy sub-and super-solutions to this equation.
Abstract: In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.
TL;DR: Non-linear ADER schemes for time-dependent scalar linear and non-linear conservation laws in one-, two- and three-space dimensions are developed and essentially non-oscillatory results are obtained for solutions with discontinuities.
TL;DR: In this paper, the mean field dynamics of the BCS paired state with a large, but finite number of Cooper pairs in the nonadiabatic regime was studied and the problem was reduced to a classical integrable Hamiltonian system.
Abstract: We obtain a complete solution for the mean-field dynamics of the BCS paired state with a large, but finite number of Cooper pairs in the nonadiabatic regime. We show that the problem reduces to a classical integrable Hamiltonian system and derive a complete set of its integrals of motion. The condensate exhibits irregular multifrequency oscillations ergodically exploring the part of the phase space allowed by the conservation laws. In the thermodynamic limit, however, the system can asymptotically reach a steady state.
TL;DR: Numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity.
Abstract: The Keller--Segel model is the classical model for chemotaxis of cell populations. It consists of a drift-diffusion equation for the cell density coupled to an equation for the chemoattractant. Here a variant of this model is studied in one-dimensional position space, where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi-stationary balance of reaction and diffusion with production of the chemoattractant by the cells. The case of small cell diffusivity is studied by asymptotic and numerical methods. On a time scale characteristic for the convective effects, convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven. Numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity. Close to each of these p...
TL;DR: A Fourier type analysis on both Petrov–Galerkin methods when solving linear one-dimensional conservation laws is performed and a comparison between the two methods is given in terms of accuracy, stability, and convergence.