Showing papers on "Conservation law published in 1996"

A fast marching level set method for monotonically advancing fronts

Abstract: A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

Numerical Approximation of Hyperbolic Systems of Conservation Laws

Abstract: From the contents: Introduction: Definitions and Examples.- Nonlinear hyperbolic systems in one space dimension.- Gas dynamics and reaction flows.- Finite Difference Schemes for one-dimensional systems.- The case of multidimensional systems.- An Introduction to Boundary conditions.

Multiple Scale and Singular Perturbation Methods

Abstract: 1. Introduction.- 1.1. Order Symbols, Uniformity.- 1.2. Asymptotic Expansion of a Given Function.- 1.3. Regular Expansions for Ordinary and Partial Differential Equations.- References.- 2. Limit Process Expansions for Ordinary Differential Equations.- 2.1. The Linear Oscillator.- 2.2. Linear Singular Perturbation Problems with Variable Coefficients.- 2.3. Model Nonlinear Example for Singular Perturbations.- 2.4. Singular Boundary Problems.- 2.5. Higher-Order Example: Beam String.- References.- 3. Limit Process Expansions for Partial Differential Equations.- 3.1. Limit Process Expansions for Second-Order Partial Differential Equations.- 3.2. Boundary-Layer Theory in Viscous, Incompressible Flow.- 3.3. Singular Boundary Problems.- References.- 4. The Method of Multiple Scales for Ordinary Differential Equations.- 4.1. Method of Strained Coordinates for Periodic Solutions.- 4.2. Two Scale Expansions for the Weakly Nonlinear Autonomous Oscillator.- 4.3. Multiple-Scale Expansions for General Weakly Nonlinear Oscillators.- 4.4. Two-Scale Expansions for Strictly Nonlinear Oscillators.- 4.5. Multiple-Scale Expansions for Systems of First-Order Equations in Standard Form.- References.- 5. Near-Identity Averaging Transformations: Transient and Sustained Resonance.- 5.1. General Systems in Standard Form: Nonresonant Solutions.- 5.2. Hamiltonian System in Standard Form Nonresonant Solutions.- 5.3. Order Reduction and Global Adiabatic Invariants for Solutions in Resonance.- 5.4. Prescribed Frequency Variations, Transient Resonance.- 5.5. Frequencies that Depend on the Actions, Transient or Sustained Resonance.- References.- 6. Multiple-Scale Expansions for Partial Differential Equations.- 6.1. Nearly Periodic Waves.- 6.2. Weakly Nonlinear Conservation Laws.- 6.3. Multiple-Scale Homogenization.- References.

Nonholonomic Mechanical Systems with Symmetry

Abstract: This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

A well-balanced scheme for the numerical processing of source terms in hyperbolic equations

Abstract: In a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable. Typical of these are gravity-driven flows such as those described by the shallow water equations over a nonuniform ocean bottom. \[ (1.10)\qquad h_t + (hu)_x = 0{\text{ and }}(hu)_t + \left( {hu^2 + \frac{{gh^2 }} {2}} \right)_x + ga_x (x)h = 0; \] Many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces. We propose here a numerical scheme, adapted to a scalar conservation law, that preserves this balance and that can, it is hoped, be extended to more general hyperbolic systems. The proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported.

Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

Abstract: We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

Convergence to equilibrium for the relaxation approximations of conservation laws

Abstract: We study the Cauchy problem for 2 × 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. © 1996 John Wiley & Sons, Inc.

The role of symmetry in fundamental physics.

Abstract: Until the 20th century principles of symmetry played little conscious role in theoretical physics. The Greeks and others were fascinated by the symmetries of objects and believed that these would be mirrored in the structure of nature. Even Kepler attempted to impose his notions of symmetry on the motion of the planets. Newton’s laws of mechanics embodied symmetry principles, notably the principle of equivalence of inertial frames, or Galilean invariance. These symmetries implied conservation laws. Although these conservation laws, especially those of momentum and energy, were regarded to be of fundamental importance, these were regarded as consequences of the dynamical laws of nature rather than as consequences of the symmetries that underlay these laws. Maxwell’s equations, formulated in 1865, embodied both Lorentz invariance and gauge invariance. But these symmetries of electrodynamics were not fully appreciated for over 40 years or more.

Hybrid Multifluid Algorithms

Abstract: Extensions of many successful single-component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components. A new algorithm based on the compressible Euler equations for multicomponent fluids augmented by the pressure evolution equation is presented. The extended set of equations offers two alternative ways to update the pressure field: (i) using the equation of state or (ii) using the pressure evolution equation. In a numerical implementation, these two procedures generally yield different answers. The former is a standard conservative update, but may produce oscillations near material interfaces; the latter is nonconservative, but becomes exact near interfaces and automatically maintains pressure equilibrium. A hybrid scheme which selects from the two pressure update procedures is presented. The scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite (very small) number of grid cells. Computed solutions exhibit oscillation-free interfaces and have {\em negligible} relative conservation errors in total energy even for very strong shocks. The proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend one's favourite scheme to solve multifluid dynamics.

Initial-boundary value problem for a scalar conservation law

Abstract: Nous etudions le probleme mixte pour l'equation ∂ t u + div [f (u)] - eΔu = 0 en (0, T) x Ω quand e tend vers zero.

A conservation law with point source and discontinuous flux function modelling continuous sedimentation

Abstract: Continuous sedimentation of solid particles in a liquid takes place in a clarifier-thickener unit, which has one feed inlet and two outlets. The aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions. The conservation of mass yields the scalar conservation law $\frac{\partial u( {x,t} )}{\partial t} + \frac{\partial }{\partial x}( {F( {u( {x,t} ),x} )} ) = s( t )\delta ( x )$, where $\delta $ is the Dirac measure, s is a source, and F is a flux function, which is discontinuous at three points in the one-dimensional space coordinate x. Under certain regularity assumptions a procedure for constructing a solution, locally in time, is presented. The nonlinear phenomena are complicated, and so is the general construction of a solution. The problem of nonuniqueness due to the discontinuities of $F( {u, \cdot } )$ is handled by a generalized entropy condition. An advantage of this approach is that the a priori boundary conditions (at the discontinuities ...

Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I

Abstract: This is the first paper in a series in which a class of nonoscillatory high order accurate self-similar local maximum principle satisfying (in scalar conservation law) shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed. In this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented, using scalar one-dimensional initial value problems as a model. For this model, the schemes are made to satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modifications, imposed as needed. The first enforces a local maximum principle; the second guarantees that no new extrema develop. The schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [Math. Comp., 49 (1987), pp. 105–121, Math. Comp...

Eckardt frame theory of interacting electrons in quantum dots

Abstract: A transformation to a moving frame (the Eckardt frame) is used to study the quantum states of interacting electrons in parabolic quantum dots in the presence of a perpendicular magnetic field. The approach is motivated by examining ground-state pair-correlation functions obtained by exact diagonalization. The main results concern the physical nature of the electron states and the origin of magic numbers. Some of the states are found to be localized about a single minimum of the potential energy. They have well-defined symmetry and are physically analogous to molecules. They are treated approximately by antisymmetrizing Eckardt frame rotational-vibrational states. This approach leads to selection rules that predict all the magic angular momentum and spin combinations found in previous numerical work. In addition, it enables the ground-state energy and low-lying excitations of the molecular states to be calculated to high accuracy. Analytic results for three electrons agree very well with the results of exact diagonalization. States that are not localized about a single minimum are also studied. They do not have distinct spatial symmetry and occur only when selection rules and conservation laws allow tunneling between states localized on different minima. These states appear to be small system precursors of fractional quantum Hall liquids. \textcopyright{} 1996 The American Physical Society.

Existence, uniqueness and mass conservation for the coagulation-fragmentation equation

Abstract: We prove global existence and uniqueness to the initial value problem for the coagulation fragmentation equation for an unbounded coagulation kernel with possible linear growth at infinity and a fragmentation kernel from a very large class of unbounded functions. We show that the solutions satisfy the mass conservation law.

Error estimates for finite element methods for scalar conservation laws

Abstract: In this paper, new a posteriors error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least $h^{{1 / 8}} $ and $h^{{1 / 8}} $, respectively, in the $L^\infty (L^1 )$-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.

Asymptotic Conservation Laws in Classical Field Theory

Abstract: A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the Arnowitt-Deser-Misner energy in general relativity. {copyright} {ital 1996 The American Physical Society.}

A microscopic model for the burgers equation and longest increasing subsequences

Abstract: We introduce an interacting random process related to Ulam's problem, or finding the limit of the normalized longest increasing subsequence of a random permutation. The process describes the evolution of a configuration of sticks on the sites of the one-dimensional integer lattice. Our main result is a hydrodynamic scaling limit: The empirical stick profile converges to a weak solution of the inviscid Burgers equation under a scaling of lattice space and time. The stick process is also an alternative view of Hammersley's particle system that Aldous and Diaconis used to give a new solution to Ulam's problem. Along the way to the scaling limit we produce another independent solution to this question. The heart of the proof is that individual paths of the stochastic process evolve under a semigroup action which under the scaling turns into the corresponding action for the Burgers equation, known as the Lax formula. In a separate appendix we use the Lax formula to give an existence and uniqueness proof for scalar conservation laws with initial data given by a Radon measure.

Hydrodynamics and Electrohydrodynamics of Liquid Crystals

Abstract: We present the hydrodynamic and electrohydrodynamic equations for uniaxial nematic liquid crystals and explain their derivation in detail. To derive hydrodynamic equations, which are valid for sufficiently small frequencies in the limit of long wavelengths, one identifies first the hydrodynamic variables, which come in two groups: quantities obeying conservation laws and variables associated with spontaneously broken continuous symmetries. As variables that characterize the spontaneously broken continuous rotational symmetries of a nematic liquid crystal we have the deviations from the preferred direction, which is characterized by the director, a unit vector that does not distinguish between head and tail.

Convergence of relaxation schemes for conservation laws

Abstract: We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.

Stability of rarefaction waves in viscous media

Abstract: We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w 0 with small mass and localized as w 0(x)= $$\mathcal{O}(|x|^{ - 1} )$$ The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error. This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass $$\mathcal{O}$$ (log (t)). These “diffusion waves” have amplitude $$\mathcal{O}$$ (t -1/2logt) in linear degenerate transversal fields and $$\mathcal{O}$$ (t -1/2logt) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.

A MUSCL method satisfying all the numerical entropy inequalities

Abstract: We consider here second-order finite volume methods for one-dimensional scalar conservation laws. We give a method to determine a slope reconstruction satisfying all the exact numerical entropy inequalities. It avoids inhomogeneous slope limitations and, at least, gives a convergence rate of Δx l/2 . It is obtained by a theory of second-order entropic projections involving values at the nodes of the grid and a variant of error estimates, which also gives new results for the first-order Engquist-Osher scheme.