Showing papers on "Conservation law published in 1995"

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

Abstract: We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.

Phase transitions in warm, asymmetric nuclear matter.

Abstract: A relativistic mean-field model of nuclear matter with arbitrary proton fraction is studied at finite temperature. An analysis is performed of the liquid-gas phase transition in a system with two conserved charges (baryon number and isospin) using the stability conditions on the free energy, the conservation laws, and Gibbs' criteria for phase equilibrium. For a binary system with two phases, the coexistence surface (binodal) is two dimensional. The Maxwell construction through the phase-separation region is discussed, and it is shown that the stable configuration can be determined uniquely at every density. Moreover, because of the greater dimensionality of the binodal surface, the liquid-gas phase transition is continuous (second order by Ehrenfest's definition), rather than discontinuous (first order), as in familiar one-component systems. Using a mean-field equation of state calibrated to the properties of nuclear matter and finite nuclei, various phase-separation scenarios are considered. The model is then applied to the liquid-gas phase transition that may occur in the warm, dilute matter produced in energetic heavy-ion collisions. In asymmetric matter, instabilities that produce a liquid-gas phase separation arise from fluctuations in the proton concentration (chemical instability), rather than from fluctuations in the baryon density (mechanical instability).

Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms

Abstract: The long-term dynamic response of non-linear geometrically exact rods under-going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time-stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy–momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.

Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force

Abstract: The spherical polar components of the Coriolis force consist of terms in sin ϕ and terms in cos ϕ, where ϕ is latitude (referred to the frame-rotation vector as polar axis). The cos ϕ Coriolis terms are not retained in the usual hydrostatic primitive equations of numerical weather prediction and climate simulation, their neglect being consistent with the shallow-atmosphere approximation and the simultaneous exclusion of various small metric terms. Scale analysis for diabatically driven, synoptic-scale motion in the tropics, and for planetary-scale motion, suggests that the cos ϕ Coriolis terms may attain magnitudes of order 10% of those of key terms in the hydrostatic primitive equations. It is argued that the cos ϕ Coriolis terms should be included in global simulation models. A global, quasi-hydrostatic model having a complete representation of the Coriolis force is proposed. Conservation of axial angular momentum and potential vorticity, as well as energy, is achieved by a formulation in which all metric terms are retained and the shallow-atmosphere approximation is relaxed. Distance from the centre of the earth is replaced by a pseudo-radius which is a function of pressure only. This model is put forward as a more accurate alternative to the traditional hydrostatic primitive equations; it preserves the desired conservation laws and may be integrated by broadly similar grid-point methods.

Convergence of the discontinuous galerkin finite element method for hyperbolic conservation laws

Abstract: We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior

Abstract: Existence and uniqueness is proved, in the class of functions satisfying a wave entropy condition, of weak solutions to a conservation law with a flux function that may depend discontinuously on th...

The semigroup generated by 2 × 2 conservation laws

Abstract: Consider the Cauchy problem for a strictly hyperbolic 2×2 system of conservation laws in one space dimension: {ie1-01} assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. This paper develops a new algorithm, based on wave-front tracking, which yields a Cauchy sequence of approximate solutions, converging to a unique limit depending continuously on the initial data. The solutions that we obtain constitute a semigroup S, defined on a set {ie1-02} of integrable functions with small total variation. For some Lipschitz constant L, we have the estimate {ie1-03}

The unique limit of the Glimm scheme

Abstract: We introduce the definitions of a standard Riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws. For a class including general 2×2 systems, it is proved that the solutions obtained by a wavefront tracking algorithm or by the Glimm scheme are precisely the semigroup trajectories. In particular, these solutions are unique and depend Lipschitz continuously on the initial data in the L1 norm.

Adaptive finite element methods for conservation laws based on a posteriori error estimates

Abstract: We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way. ©1995 John Wiley & Sons, Inc.

On scalar conservation laws with point source and discontinuous flux function

Abstract: The conservation law studied is partial derivative u(x,t)/partial derivative t + partial derivative/partial derivative x (F(u(x,t),x)) = s(t)delta(x), where u is a concentration, s is a source, delta is the Dirac measure, and is the flux function. The special feature of this problem is the discontinuity that appears along the t-axis and the curves of discontinuity that go into and emanate from it. Necessary conditions for the existence of La piecewise smooth solution are given. Under some regularity assumptions sufficient conditions are given enabling construction of piecewise smooth solutions by the method of characteristics. The selection of a unique solution is made by a coupling condition at x = 0, which is a generalization of the classical entropy condition and is justified by studying a discretized version of the problem by Godunov's method. The motivation for studying this problem is the fact that it arises in the modelling of continuous sedimentation of solid particles in a liquid. (Less)

Positive schemes and shock modelling for compressible flows

Abstract: SUMMARY A unified theory of non-oscillatory finite volume schemes for both structured and unstructured meshes is developed in two parts. In the first part, a theory of local extremum diminishing (LED) and essentially local extremum diminishing (ELED) schemes is developed for scalar conservation laws. This leads to symmetric and upstream limited positive (SLIP and USLIP) schemes which can be formulated on either structured or unstructured meshes. The second part examines the application of similar ideas to the treatment of systems of conservation laws. An analysis of discrete shock structure leads to conditions on the numerical flux such that stationary discrete shocks can contain a single interior point. The simplest formulation which meets these conditions is a convective upwind and split pressure (CUSP) scheme, in which the coefficient of the pressure differences is fully determined by the coefficient of convective diffusion. Numerical results are presmted which confirm the properties of these schemes. This paper presents a unified formulation of non-oscillatory discretization schemes for the calculation of compressible flows on both structured and unstructured meshes. Over the past decade the principles underlying the design of non-oscillatory discretization schemes have been quite well established, and numerous variations of artificial diffusion, upwind biasing and flux splitting have been proposed and tested.' - * The non-oscillatory properties of the schemes analysed here are secured through the introduction of artificial viscosity which produces an upwind bias. This exactly reproduces an upwind scheme when the minimum sufficient amount of viscosity is used. Higher-order accuracy is obtained by the use of higher-order diffusive terms, with limiters to preserve monotonicity constraints. Schemes which blend low and high-order diffusion,' and both symmetric and upstream constructions using anti-diffusive terms controlled by limiters,' are readily included within the framework of this paper. Two main issues arise in the design of non-oscillatory discrete schemes. First there is the issue of how to construct an approximation to a scalar convection or convection-diffusion equation which is non-oscillatory, captures discontinuities with high resolution, and is sufficiently accurate. Second there is the issue of how to construct a numerical flux for a system of equations with waves travelling at different speeds, and sometimes in opposite directions. These two issues can be treated essentially independently, and by combining alternative non-oscillatory formulations with different constructions of the numerical flux one arrives at a matrix of candidate high

Convergence of the finite volume method for multidimensional conservation laws

Abstract: We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations (“flat elements” are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169–210 and J. Numer. Anal., 30 (1993), pp. 675–700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna’s uniqueness result in the class of entropy measure-valued solutions.

A variational calculus for discontinuous solutions of systems of conservation laws

Abstract: (1995). A variational calculus for discontinuous solutions of systems of conservation laws. Communications in Partial Differential Equations: Vol. 20, No. 9-10, pp. 1491-1552.

Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations

Abstract: This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher-order schemes (filtered to ``preserve'' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians.

Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

Abstract: Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.

For Every Generalization Action, Is There Really an Equal and Opposite Reaction? Analysis of the Conservation Law for Generalization Performance

Abstract: The “Conservation Law for Generalization Performance” [Schaffer, 1994] states that for any learning algorithm and bias, “generalization is a zero-sum enterprise.” In this paper we study the law and show that while the law is true, the manner in which the Conservation Law adds up generalization performance over all target concepts, without regard to the probability with which each concept occurs, is relevant only in a uniformly random universe. We then introduce a more meaningful measure of generalization, expected generalization performance. Unlike the Conservation Law's measure of generalization performance (which is, in essence, defined to be zero), expected generalization performance is conserved only when certain symmetric properties hold in our universe. There is no reason to believe, a priori, that such symmetries exist; learning algorithms may well exhibit non-zero (expected) generalization performance.

Integrable Multi-Component Hybrid Nonlinear Schrödinger Equations

Abstract: Presented is a set of integrable multi-component hybrid nonlinear Schrodinger (MCHNS) equations. Each multi-component equation is a superposition of the nonlinear Schrodinger (NS) equation and the derivative nonlinear Schrodinger (DNS) equation. For the MCHNS equations, the inverse scattering formulation is given. The gauge transformation relating 2-component hybrid nonlinear Schrodinger equation with the Manakov equation is explicitly shown. This also confirms that the former is integrable since the latter is integrable.