Global Conservation Laws Research Articles

Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes

This paper follows up on the author's recent paper Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthalpy. In the case of the gas dynamics equations the previous formulation leads to an entropy preserving (EP) scheme. It is shown in the present paper that it is also possible to construct the flux of a conservative finite volume scheme to produce a kinetic energy preserving (KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A proof is presented for three dimensional discretization on arbitrary grids. Both the EP and KEP schemes have been applied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The computations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme performed better in the tests.

Non-Local Scattering Kernel and the Hydrodynamic Limit

In this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory. We consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall. This results in a scattering kernel which is a non-local generalization of the classical Maxwell scattering kernel. The proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation. We study the hydrodynamic limit performing a Knudsen layer analysis, and derive a new class of (weakly) nonlocal boundary conditions to be imposed to the Navier–Stokes equations.

On the low-frequency limit of flame transfer functions

The response of a premixed flame to quasi-steady flow perturbations is considered. It is found that in this low-frequency limit, constraints on the flame transfer function can be established from global conservation laws for mass, energy, and momentum. For example, the transfer function between velocity fluctuations and heat release of a perfectly premixed flame without fluctuations of equivalence ratio should be unity in the limit of zero frequency, while for a combustion system with constant mass flow rate of fuel, the transfer function should tend toward zero. It is demonstrated that these considerations can be employed to reduce the number of unknowns in analytical flame models or identify invalid modeling assumptions.

Global conservation laws and femtoscopy of small systems

It is increasingly important to understand, in detail, two-pion correlations measured in p+p and d+A collisions. In particular, one wishes to understand the femtoscopic correlations, in order to compare to similar measurements in heavy ion collisions. However, in the low-multiplicity final states of these systems, global conservation laws generate significant N-body correlations which project onto the two-pion space in non-trivial ways and complicate the femtoscopic analysis. We discuss a model-independent formalism to calculate and account for these correlations in measurements.

Mass and momentum conservation in the simplified flood routing models

Summary In the paper the conservative properties of the simplified flood routing models in form of the advective or advective–diffusive transport equation are analysed. The aim of the analysis is to explain the reasons of the mass balance error, which is observed in the numerical solutions of both non-linear kinematic and diffusive wave equations. It appeared that depending on the assumed conservative form of the kinematic wave equation, this model satisfies either the global mass conservation law or the global momentum conservation law. Both laws are satisfied simultaneously by the linear equation only. Moreover it is shown that for the non-linear diffusive wave model with the flow discharge Q ( x , t ) as unknown, the mass and momentum balance errors cannot be eliminated. This is caused by the presence of a diffusion term in the non-divergent form, which arises during the derivation the non-linear diffusive wave model.

Multi-symplectic Runge–Kutta–Nyström methods for nonlinear Schrödinger equations with variable coefficients

In this paper, we consider Runge–Kutta–Nyström (RKN) methods applied to nonlinear Schrödinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nyström methods in spatial direction and symplectic Runge–Kutta methods in temporal direction for NLSEvc leads to multi-symplectic integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. It is shown that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. We present a (4-order) multi-symplectic RKN method and use it in numerical simulation of quasi-periodically solitary waves for NLSEvc, and we compare the multi-symplectic RKN method with a non-multi-symplectic RKN method on the errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge. The precise conservation of discrete charge under the multi-symplectic RKN discretizations is attested numerically. Some numerical superiorities of the multi-symplectic RKN methods are revealed.

A note on multisymplectic Fourier pseudospectral discretization for the nonlinear Schrödinger equation

Based on the multisymplectic Fourier pseudospectral scheme for the nonlinear Schrödinger (NLS) equation, we investigate some discrete properties corresponding to local conservation laws of the original equation. The discrete normal conservation law is proved, and the error estimation of local and global energy conservation laws are also obtained. Numerical experiments for cubic NLS equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results.

Review of Hydrology: An Introduction by Wilfried BrutsaertHydrology: An IntroductionCambridge University Press

As stated by the author, the goal of this book is “to present a coherent introduction to some of the concepts and relationships needed to describe the distribution and transport of water in the natural environment.” The book admirably accomplishes this goal through the wealth of information contained in 14 chapters and one appendix. Chapter 1 furnishes an excellent introduction to hydrology; the remaining chapters are divided into four parts: Part I covers water in the atmosphere, Part II covers water on the surface, Part III covers water below the surface, and Part IV covers flows at the catchment scale in response to precipitation. After beginning with the definition and scope of hydrology, Chapter 1 discusses the hydrologic cycle, the global water balance, methodologies, and procedures and conservation laws. The discussion is lucid and informative. Since this chapter is introductory, it would have been desirable to provide a broader discussion of hydrology—including water quality, erosion and sedimentation, land use effects, snow and glaciers, ecosystems, design considerations, and measurements—and then limit the scope to what is treated in the book. Part I—on water in the atmosphere—includes three chapters. Chapter 2 succinctly presents elements of fluid mechanics of the lower atmosphere—including water vapor, atmospheric stability, turbulent transport, atmospheric boundary layer, turbulence similarity, and energy budget constraints. The discussion is excellent and is much better than what is presented in most hydrology textbooks. Chapter 3 discusses precipitation. It begins with a discussion of the formation of precipitation, including cooling of the air, moisture supply, water recycling, and types of precipitation. The chapter then discusses major precipitation weather systems— including extratropical cyclones and fronts, extratropical convective weather systems, seasonal tropical systems, large-scale tropical convective systems, and orographic effects; precipitation distribution on the ground, including spatial distribution, temporal distribution, and runoff design rainfall data; and interception, including interception loss mechanisms, vegetation structural parameters, and empirical equations. The chapter concludes with a discussion of operational precipitation measurements. This chapter is excellent and comprehensive. Chapter 4 deals with evaporation. It discusses evaporation mechanisms, mass transfer formulations, energy budget and related formulations, water budget methods, and evaporation climatology. The entire discussion is detailed and well illustrated with graphs and tables. Part I of the book stands out, especially when compared with other hydrology textbooks. It is rich in information, and the discussion is very appealing.

Traveling ion channel density waves affected by a conservation law

A model of mobile, charged ion channels embedded in a biomembrane is investigated. The ion channels fluctuate between an opened and a closed state according to a simple two-state reaction scheme whereas the total number of ion channels is a conserved quantity. Local transport mechanisms suggest that the ion channel densities are governed by electrodiffusionlike equations that have to be supplemented by a cable-type equation describing the dynamics of the transmembrane voltage. It is shown that the homogeneous distribution of ion channels may become unstable to either a stationary or an oscillatory instability. The nonlinear behavior immediately above threshold of an oscillatory bifurcation occurring at finite wave number is analyzed in terms of amplitude equations. Due to the conservation law imposed on ion channels, large-scale modes couple to the finite-wave-number instability and have thus to be included in the asymptotic analysis near the onset of pattern formation. A modified Ginzburg-Landau equation extended by long-wavelength stationary excitations is established, and it is highlighted how the global conservation law affects the stability of traveling ion channel density waves.

On point blast theory

The problem of a spherical shock front generated in a point explosion and traveling in a homogeneous medium is analytically studied with account for counterpressure on the entire infinite interval of its existence. For this purpose, asymptotic representations of the excess pressure in the shock wave near and far away from the energy release point are matched. It is possible analytically to continue the four-term expansion for the far zone involving unknown constants, so that it rigorously coincides with the four-term power expansion of the solution for the singular point, that is, the blast center. The problem of determination of the unknown constants is mathematically closed by the derived entropy loss integral which expresses the global energy conservation law. The analytical dependence of the excess pressure in the shock wave on distance thus obtained is in good agreement with the results of numerical calculations.

Gravitational Radiation Reaction and Inspiral Waveforms in the Adiabatic Limit

We describe progress evolving an important limit of binaries in general relativity: stellar mass compact objects spiraling into much larger black holes. Such systems are of great observational interest. We have developed tools to compute for the first time the radiation from generic orbits. Using global conservation laws, we find the orbital evolution and waveforms for special cases. For generic orbits, inspirals and waveforms can be found by augmenting our approach with an adiabatic self-force rule due to Mino. Such waveforms should be accurate enough for gravitational-wave searches.

Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm

We present the critical theory of a number of zero temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S = 1/2 square lattice antiferromagnet between the Neel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second order quantum phase transition. The critical theory is not expressed in terms of the order parameters characterizing either state (as would be the case in Landau-Ginzburg-Wilson theory) but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half-filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z_2 spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.

Exact solutions of exactly integrable quantum chains by a matrix product ansatz

Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz, the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian defines the algebraic properties of the matrices defining the amplitudes. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. The matrix product ansatz we propose allows an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t–J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with z-magnetic fields at the surface and arbitrary hard-core exclusion among the spins.

Quantum coherence in the presence of unobservable quantities

State representations summarize our knowledge about a system. When unobservable quantities are introduced the state representation is typically no longer unique. However, this non-uniqueness does not affect subsequent inferences based on any observable data. We demonstrate that the inference-free subspace may be extracted whenever the quantity's unobservability is guaranteed by a global conservation law. This result can generalize even without such a guarantee. In particular, we examine the coherent-state representation of a laser where the absolute phase of the electromagnetic field is believed to be unobservable. We show that experimental coherent states may be separated from the inference-free subspaces induced by this unobservable phase. These physical states may then be approximated by coherent states in a relative-phase Hilbert space.

A model for diffusion-controlled solidification of ternary alloys in mushy layers With an Appendix by A. F. Thompson, H. E. Huppert M. G. Worster.

We describe a model for non-convecting diffusion-controlled solidification of a ternary (three-component) alloy cooled from below at a planar boundary. The modelling extends previous theory for binary alloy solidification by including a conservation equation for the additional solute component and coupling the conservation equations for heat and species to equilibrium relations from the ternary phase diagram. We focus on growth conditions under which the solidification path (liquid line of descent) through the ternary phase diagram gives rise to two distinct mushy layers. A primary mushy layer, which corresponds to solidification along a liquidus surface in the ternary phase diagram, forms above a secondary (or cotectic) mushy layer, which corresponds to solidification along a cotectic line in the ternary phase diagram. These two mushy layers are bounded above by a liquid layer and below by a eutectic solid layer. We obtain a one-dimensional similarity solution and investigate numerically the role of the control parameters in the growth characteristics. In the special case of zero solute diffusion and zero latent heat an analytical solution can be obtained. We compare our predictions with previous experimental results and with theoretical results from a related model based on global conservation laws described in the Appendix. Finally, we discuss the potentially rich convective behaviour anticipated for other growth conditions.

A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law

One-dimensional, second-order finite-difference approximations of the derivative are constructed which satisfy a global conservation law. Creating a second-order approximation away from the boundary is simple, but obtaining appropriate behavior near the boundary is difficult, even in one dimension on a uniform grid. In this article we exhibit techniques that allow the construction of discrete versions of the divergence and gradient operator that have high-order approximations at the boundary. We construct such discretizations in the one-dimensional situation which have fourth-order approximation both on the boundary and in the interior. The precision of the high-order mimetic schemes in this article is as high as possible at the boundary points (with respect to the bandwidth parameter). This guarantees an overall high order of accuracy. Furthermore, the method described for the calculation of the approximations uses matrix analysis to streamline the various mimetic conditions. This contributes to a marked clarity with respect to earlier approaches. This is a crucial preliminary step in creating higher-order approximations of the divergence and gradient for nonuniform grids in higher dimensions.

Dynamical surface structures in multiparticle-correlated surface growths.

We investigate the scaling properties of the interface fluctuation width for the Q-mer and Q-particle-correlated deposition-evaporation models. These models are constrained with a global conservation law that the particle number at each height is conserved modulo Q. In equilibrium, the stationary roughness is anomalous but universal with the roughness exponent alpha=1/3, while the early time evolution shows nonuniversal behavior with the growth exponent beta varying with models and Q. Nonequilibrium surfaces display diverse growing and stationary behaviors. The Q-mer model shows a faceted structure, while the Q-particle-correlated model shows a macroscopically grooved structure.

Phase ordering with a global conservation law: Ostwald ripening and coalescence.

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Third family of superdense stars in the presence of antikaon condensates

The formation of ${K}^{\ensuremath{-}}$ and ${K}^{0}$ condensation in $\ensuremath{\beta}$-equilibrated hyperonic matter is investigated within a relativistic mean field model. In this model, baryon-baryon and (anti)kaon-baryon interactions are mediated by the exchange of mesons. It is found that antikaon condensation is not only sensitive to the equation of state but also to the antikaon optical potential depth. For large values of the antikaon optical potential depth, ${K}^{\ensuremath{-}}$ condensation sets in before the appearance of negatively charged hyperons. We treat ${K}^{\ensuremath{-}}$ condensation as a first-order phase transition. The Gibbs criteria and global charge conservation laws are used to describe the mixed phase. Nucleons and $\ensuremath{\Lambda}$ hyperons behave dynamically in the mixed phase. A second-order phase transition to ${K}^{0}$ condensation occurs in the pure ${K}^{\ensuremath{-}}$ condensed phase. Along with ${K}^{\ensuremath{-}}$ condensation, ${K}^{0}$ condensation makes the equation of state softer, thus resulting in smaller maximum mass stars compared with the case without any condensate. This equation of state also leads to a stable sequence of compact stars called the third family branch, beyond the neutron star branch. The compact stars in the third family branch have different compositions and smaller radii than that of the neutron star branch.

Geometric Integrators for the Nonlinear Schrödinger Equation

Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduced for solving systems possessing a certain multisymplectic structure. Some of the characteristic features of the method are its local nature (independent of boundary conditions) and an equal treatment of spatial and temporal variables. The nonlinear Schrödinger equation (NLS) has a multisymplectic formulation, and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the NLS. In the numerical experiments, the multisymplectic concatenated midpoint scheme (a centered cell discretization) is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff. On the other hand, an integrable Hamiltonian semi-discretization of NLS from Ablowitz and Ladik (AL) possesses a full set of global conservation laws and a noncanonical symplectic structure. We generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the AL type. Another approach examined in the paper is the introduction of transformations to reduce the AL system to either (1) separable form or (2) canonical form and then apply standard schemes in the new coordinates. All of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions. The performance of the schemes as well as interrelations among various geometric features are discussed.