Showing papers on "Conservation law published in 2004"

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

Abstract: We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources

Abstract: Introduction.- 1. Quasilinear systems and conservation laws.- 2. Conservative schemes.- 3. Source terms.- 4. Nonconservative schemes.- 5. Multidimensional finite volumes with sources.- 6. Numerical test with source.- Bibliography

Swarming patterns in a two-dimensional kinematic model for biological groups ∗

Abstract: We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree o...

A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws

Abstract: We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel

Geometrically Exact, Intrinsic Theory for Dynamics of Curved and Twisted Anisotropic Beams

Abstract: A formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams. When the applied loads at the ends of, and distributed along, the beam are independent of the deformation, neither displacement nor rotation variables appear: an intrinsic formulation. Like well-known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams, the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two. Advantages of such a formulation are demonstrated with a simple example. When the initial curvature and twist are constant along the beam, two space-time conservation laws are shown to exist, one being a work-energy relation and the other a generalized impulse-momentum relation. These laws can be used, for example, as benchmarks to check the accuracy of any proposed solution, including time-marching and finite element schemes. The structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization. A particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end-loaded beams that, by virtue of the Kirchhoff analogy, leads to a time-marching scheme for the dynamics of a pivoted rigid body in a gravity field. This time-marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy, whereas the analogous spatial discretization scheme for the nonlinear static behavior of end-loaded beams satisfies analogous integrals of deformation along the beam span. Remarkably, a straightforward generalization of these discretization schemes is shown to satisfy both space-time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space-time element.

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Abstract: A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of ‘sufficiently many’ n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.

Entropy and Global Existence for Hyperbolic Balance Laws

Abstract: This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchut’s discrete velocity BGK models approximating hyperbolic systems of conservation laws.

Finite Volume Methods: Foundation and Analysis

Abstract: Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form. This article reviews elements of the foundation and analysis of modern finite volume methods. The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, non-oscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of diffusion terms and the extension to systems of nonlinear conservation laws.

On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems

Abstract: This paper is concerned with the numerical approximations of Cauchy problems for one- dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well- balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by (Toumi, J. Comp. Phys. 102 (1992) 360-373). Next, this general theory is applied to obtain well- balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by (Bermudez and Vazquez-Cendon, Comput. Fluids 23 (1994) 1049-1071); in the case of two layer flows, they are compared with the numerical scheme presented by (Castro, Mac´ias and Pares, ESAIM: M2AN 35 (2001) 107-127).

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids III: One Dimensional Systems and Partition Optimization

Abstract: In this paper, the third in a series, the Spectral Volume (SV) method is extended to one-dimensional systems—the quasi-1D Euler equations. In addition, several new partitions are identified which optimize a certain form of the Lebesgue constant, and the performance of these partitions is assessed with the linear wave equation. A major focus of this paper is to verify that the SV method is capable of achieving high-order accuracy for hyperbolic systems of conservation laws. Both steady state and time accurate problems are used to demonstrate the overall capability of the SV method.

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

Abstract: A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG)[1] and the Spectral Volume (SV)[2] methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference (FD)[3] and finite-volume (FV)[4] methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for non-linear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly.

Convergence of the lax-friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux

Abstract: The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form where the coefficient k(x,t) is allowed to be discontinuous along curves in the (x,t) plane. In contrast to most of the existing literature on problems with discontinuous coefficients, here the convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of k(x,t). Following [14], the authors propose a definition of entropy solution that extends the classical Kružkov definition to the situation where k(x,t) is piecewise Lipschitz continuous in the (x,t) plane, and prove the stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x,t) is discontinuous. It is shown that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.

The Riemann problem for a class of resonant hyperbolic systems of balance laws

Abstract: We solve the Riemann problem for a class of resonant hyperbolic systems of balance laws The systems are not strictly hyperbolic and the solutions take their values in a neighborhood of a state where two characteristic speeds coincide Our construction generalizes the ones given earlier by Isaacson and Temple for scalar equations and for conservative systems The class of systems under consideration here includes, in particular, a model from continuum physics that describes the evolution of a fluid flow in a nozzle with discontinuous cross-section

Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space

Abstract: Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is presented for two-phase flow in porous media. A consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration. Furthermore, for regular solutions, uniqueness has been shown.

Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

Abstract: We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L1 stability) of the entropy solution in the BVt class (functions W(x,t) with ∂tW being a finite measure). The existence of a BVt entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.

Hydrodynamic Structure of the Augmented Born-Infeld Equations

Abstract: The Born-Infeld system is a nonlinear version of Maxwell’s equations. We first show that, by using the energy density and the Poynting vector as additional unknown variables, the BI system can be augmented as a 10×10 system of hyperbolic conservation laws. The resulting augmented system has some similarity with magnetohydrodynamics (MHD) equations and enjoys remarkable properties (existence of a convex entropy, Galilean invariance, full linear degeneracy). In addition, the propagation speeds and the characteristic fields can be computed in a very easy way, in contrast with the original BI equations. Then, we investigate several limit regimes of the augmented BI equations, by using a relative-entropy method going back to Dafermos, and recover the Maxwell equations for low fields, some pressureless MHD equations for high fields, and pressureless gas equations for very high fields.

Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid

Abstract: In this paper, we consider a two-dimensional fluid-rigid body problem. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the dynamics of the rigid body is governed by the conservation laws of linear and angular momentum. The rigid body is supposed to be an infinite cylinder of circular cross-section. Our main result is the existence and uniqueness of global strong solutions.

A Variational Complex for Difference Equations

Abstract: An analogue of the Poincare lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler–Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler–Lagrange system).

Conservation laws for the voter model in complex networks

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.