Showing papers on "Conservation law published in 1994"

Hyperbolic conservation laws with stiff relaxation terms and entropy

Abstract: We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.

Onsager's conjecture on the energy conservation for solutions of Euler's equation

Abstract: We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.

The quantum hydrodynamic model for semiconductor devices

Abstract: The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections These quantum corrections are $O( {\hbar ^2 } )$ The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms These quantum terms allow particles to tunnel through potential barriers and to build up in potential wellsThe three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind methodSimulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v

Three-dimensional adaptive mesh refinement for hyperbolic conservation laws

Abstract: A local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described. The method is based on the use of local grid patches superimpo...

Kinetic formulation of the isentropic gas dynamics and $p$-systems

Abstract: We consider the 2 x 2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called the p-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce new L°° estimates using moments lemma and prove L°° — w* stability for 7 > 3.

Why nonconservative schemes converge to wrong solutions: error analysis

Abstract: This paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws. We show that these schemes converge strongly in LI norm to the solution of an inhomogeneous conservation law containing a Borel measure source term. Moreover, we analyze the properties of this Borel measure, and derive a sharp estimate for the L1 error between the limit function given by the scheme and the correct solution. In general, the measure source term is of the order of the entropy dissipation measure associated with the scheme. In certain cases, the error can be small for short times, which makes it difficult to detect numerically. But generically, such an error will grow in time, and this would lead to a large error for large-time calculations. Finally, we show that a local correction of any high-order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution.

A new energy and momentum conserving algorithm for the non‐linear dynamics of shells

Abstract: A numerical time-integration scheme for the dynamics of non-linear elastic shells is presented that simultaneously and independent of the time-step size inherits exactly the conservation laws of total linear, total angular momentum as well as total energy The proposed technique generalizes to non-linear shells recent work of the authors on non-linear elastodynamics and is ideally suited for long-term/large-scale simulations The algorithm is second-order accurate and can be immediately extended with no modification to a fourth-order accurate scheme The property of exact energy conservation induces a strong notion of non-linear numerical stability which manifests itself in actual simulations

On a cell entropy inequality for discontinuous Galerkin methods

Abstract: We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.

Dissipation Induced Instabilities

Abstract: The main goal of this paper is to prove that if the energy-momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group—thus, we consider internal dissipation. This also includes the special case of systems with no symmetry and ordinary equilibria. The theorem is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev-Lyapunov function, with those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. The main achievement is to strengthen Chetaev’s methods to the context of the block diagonalization version of the energy momentum method given by Lewis, Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagonal form, as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.

A Balance Condition for Stochastic Numerical Models with Application to the El Niño-Southern Oscillation

Abstract: Stochastic forcing due to unresolved processes adds energy to a measurable system. Although this energy is added randomly in time, conservation laws still apply. A balance condition for stochastically driven systems is discussed. This “fluctuation-dissipation relation” may be used either to deduce the geographical properties of the stochastic forcing from data given a model for the evolution of the macroscopic variables or to diagnose energy conservation in a stochastic numerical model. The balance condition in its first role was applied to sea surface temperatures (SSTs) in the Indo-Pacific basin. A low-dimensional empirical dynamical model of SSTs was generated in such a way that observed statistical properties of the field are preserved. Experiments varying the stochastic forcing in this model indicated how the geographical characteristics of the forcing affect the distribution of variance among the various normal modes thereby determining the dominant timescales of the SST field. These result...

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

Abstract: The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t +f(u) x =μu xx with the initial datau 0 which tend to the constant statesu ± asx→±∞. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f′(u ±). Moreover, the rate of asymptotics in time is investigated. For the casef′(u+)

Local piecewise hyperbolic reconstruction of numerical fluxes for nonlinear scalar conservation laws

Abstract: This paper constructs a local third order accurate shock capturing method for hyperbolic scalar conservation laws, based on numerical fluxes with a total variation diminishing (TVD) Runge–Kutta evolution in time, using the idea recently introduced by C. W. Shu and S. J. Osher for essentially nonoscillatory (ENO) methods. The constructed method is an upwind conservative scheme that is local in the sense that numerical fluxes are reconstructed without using extrapolation from the data of the smoothest neighboring cell. To design the method, a new concept of local smoothing is introduced to prevent the increasing of total variation of the solution near discontinuities and to achieve third order accuracy. The method becomes third order accurate in smooth regions of the solution, except at local extrema where it may degenerate to $O(h^{3/2} )$, thus giving better accuracy than TVD methods at local extrema. The main advantage of this method lies on the property that is more local than that of ENO and TVD upwind...

Formation of shape-preserving pulses in a nonlinear adiabatically integrable system

Abstract: We have obtained a class of solitary wave solutions to novel exactly integrable nonlinear wave equations. Conservation laws can be identified and velocities of propagation predicted. We propose to test our predictions in the optical domain with two-color experiments.

Nonlinearly stable compact schemes for shock calculations

Abstract: The applications of high-order, compact finite difference methods in shock calculations are discussed. The main concern is to define a local mean which will serve as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while maintaining the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total-variation stable in one space dimension and maximum-norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. These ideas can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.

Vector-matrix generalizations of classical integrable equations

Abstract: Some vector-matrix generalizations, both known and new, for well-known integrable equations are presented. All of them possess higher symmetries and conservation laws.

An error estimate for finite volume methods for multidimensional conservation laws

Abstract: In this paper, an L°°(Ll )-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ft'/4 , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con- tinuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.

Vortex laws and field line invariants in polytropic field‐aligned MHD flow

Abstract: The consequences of conservation of angular momentum in, single- or double-polytropic, steady, compressible, magnetohydrodynamic (MHD) flow parallel to the magnetic field are examined, the principal result being the MHD counterparts of Crocco's theorem and Lord Kelvin's theorem, both expressed in terms of a generalized vorticity. Under special assumptions concerning geometry and/or homogeneity of the general field line invariants, these vortex laws lead to additional field line invariants which describe the conservation of generalized angular momentum in the plasma, taking into account torques produced by j×B forces and Coriolis forces.

Energy-momentum conservation in gravity theories.

Abstract: We discuss general properties of the conservation law associated with a local symmetry. Using Noether's theorem and a generalized Belinfante symmetrization procedure in 3+1 dimensions, a symmetric energy-momentum (pseudo) tensor for the gravitational Einstein-Hilbert action is derived and discussed in detail. In 2+1 dimensions, expressions are obtained for energy and angular momentum arising in the ISO(2,1) gauge-theoretical formulation of Einstein gravity. In addition, an expression for energy in a gauge-theoretical formulation of the string-inspired (1+1)-dimensional gravity is derived and compared with the ADM definition of energy.

Numerical models of steady rolling for non‐linear viscoelastic structures in finite deformations

Abstract: A Lagrangian formulation of constitutive laws for a viscoelastic material based on irreversible thermodynamics is first presented. These laws are expressed by a non-linear differential equation governing the evolution of an internal variable. Then equations describing the steady rolling of an axisymmetric viscoelastic structure are obtained from the conservation laws of continuum mechanics. A finite element approximation and a solution technique of the algebraic system is proposed. The eiimination of the internal variable allows one to keep an elastic-like algorithm with an independent solution of the viscoelastic equation. Numerical tests show the feasibility and the efficiency of the proposed methods in large three-dimensional situations.

The structure of streamers in N2. I. fast method of space-charge dominated plasma simulation

Abstract: A novel technique of simulation of low-temperature gas-discharge plasmas when space-charge effects are dominant is described. It is shown that the Poisson equation for electric field can be replaced by a much simpler current conservation law so that the solution would automatically obey the Poisson equation. A one-dimensional version of this method was tested on the problem of cathode region formation in pre-ionized atmospheric pressure nitrogen. A two-dimensional scheme for cylindrical coordinates is described. The method proposed allows one to simulate fully the two-dimensional problem of streamer dynamics in an over-volted gap on a personal computer, as described in the companion paper.

Multidimensional discretization of conservation laws for unstructured polyhedral grids

Abstract: To the extent possible, a discretized system should satisfy the same conservation laws as the physical system. The author considers the conservation properties of a staggered-grid Lagrange formulation of the hydrodynamics equations (SGH) which is an extension of a ID scheme due to von Neumann and Richtmyer (VNR). The term staggered refers to spatial centering in which position, velocity, and kinetic energy are centered at nodes, while density, pressure, and internal energy are at cell centers. Traditional SGH formulations consider mass, volume, and momentum conservation, but tend to ignore conservation of total energy, conservation of angular momentum, and requirements for thermodynamic reversibility. The author shows that, once the mass and momentum discretizations have been specified, discretization for other quantities are dictated by the conservation laws and cannot be independently defined. The spatial discretization method employs a finite volume procedure that replaces differential operators with surface integrals. The method is appropriate for multidimensional formulations (1D, 2D, 3D) on unstructured grids formed from polygonal (2D) or polyhedral (3D) cells. Conservation equations can then be expressed in conservation form in which conserved currents are exchanged between control volumes. In addition to the surface integrals, the conservation equations include source terms derived from physical sources or geometrical considerations. In Cartesian geometry, mass and momentum are conserved identically. Discussion of volume conservation will be temporarily deferred. The author shows that the momentum equation leads to a form-preserving definition for kinetic energy and to an exactly conservative evolution equation for internal energy. Similarly, the author derives a form-preserving definition and corresponding conservation equation for a zone-centered angular momentum.

Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations

Abstract: Using the inverse strong symmetry of the Korteweg–de Vries (KdV) equation on the trivial symmetry and τ0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are expressed explicitly by the multi‐integrations of the Jost function of the KdV equation and constitute an infinite dimensional Lie algebra together with two hierarchies of the known symmetries. Contrary to the general belief, the time‐independent symmetry groups of the KdV and mKdV equations are non‐Abelian and the infinite dimensional Lie algebras of the KdV and mKdV equations are nonisomorphic though two equations are related by the Miura transformation. Starting from these sets of symmetries, four hierarchies of the integrodifferential KdV equations, which can be solved by the Schrodinger inverse scattering transformation method, are obtained. Some of these hierarchies enjoy a common strong symmetry and/or same local conserved densities.

Structure of the conservation laws in integrable spin chains with short range interactions

Abstract: We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This construction is generalized for the su(M) invariant integrable chain. We also indicate that a quantum recursive (ladder) operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. We show that in the quantum continuous limits of the XYZ model, the ladder property of the boost operator disappears. For the Hubbard model we demonstrate the non-existence of a ladder operator. Nevertheless, the general structure of the conserved charges is indicated, and the expression for the terms linear in the model's free parameter for all charges is derived in closed form.

Evolution of the Weibel instability in relativistically hot electron–positron plasmas

Abstract: Analytical and numerical studies of the evolution of the Weibel instability in relativistically hot electron–positron plasmas are presented. Appropriate perturbations on the electromagnetic fields and the particle orbits, corresponding to a single unstable mode, are determined analytically and used as initial conditions in the numerical simulations to excite a single unstable mode. A simple estimate of the saturation amplitude is also obtained analytically. Numerical simulations are carried out when a single unstable mode is favorably excited. Comparisons of the simulation results with the analytical ones show very good agreement. Also observed in the simulations are mode competition, mode suppression, and the difference in the long‐term evolution between the magnetized and unmagnetized plasmas. For relativistic unmagnetized plasmas, energy‐like global constraints, which are conservation laws in addition to the conservation of energy and momentum, are derived. Numerical simulations of the multimode evolut...