Single Conservation Law Research Articles

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

Abstract The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl -regular Nijenhuis operators. We prove the splitting theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl -regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl -regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl -regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e. its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry.

We study the Oslo model, a paradigm for absorbing-phase transition, on a one-dimensional ring of L sites with a fixed global density ρ[over ¯]; we consider the system strictly above critical density ρ_{c}. Notably, microscopic dynamics conserve both mass and center of mass (CoM), but lack time-reversal symmetry. We show that, despite having highly constrained dynamics due to CoM conservation, the system exhibits diffusive relaxation away from criticality and superdiffusive relaxation near criticality. Furthermore, the CoM conservation severely restricts particle movement, causing the mobility-a transport coefficient analogous to the conductivity for charged particles-to vanish exactly. Indeed, the steady-state temporal growth of current fluctuation is qualitatively different from that observed in diffusive systems with a single conservation law. Remarkably, far from criticality where the relative density Δ=ρ[over ¯]-ρ_{c}≫ρ_{c}, the second cumulant, or the variance, 〈Q_{i}^{2}(T,Δ)〉_{c}, of current Q_{i} across the ith bond up to time T in the steady-state saturates as 〈Q_{i}^{2}〉_{c}≃Σ_{Q}^{2}(Δ)-constT^{-1/2}; near criticality, it grows subdiffusively as 〈Q_{i}^{2}〉_{c}∼T^{α}, with 0<α<1/2, and eventually saturates to Σ_{Q}^{2}(Δ). Interestingly, the asymptotic current fluctuation Σ_{Q}^{2}(Δ) is a nonmonotonic function of Δ: It diverges as Σ_{Q}^{2}(Δ)∼Δ^{2} for Δ≫ρ_{c} and Σ_{Q}^{2}(Δ)∼Δ^{-δ}, with δ>0, for Δ→0^{+}. Using a mass-conservation principle, we exactly determine the exponents δ=2(1-1/ν_{⊥})/ν_{⊥} and α=δ/zν_{⊥} via the correlation-length and dynamic exponents, ν_{⊥} and z, respectively. Finally, we show that in the steady state the self-diffusion coefficient D_{s}(ρ[over ¯]) of tagged particles is connected to activity through the relation D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯].

Asymptotic stability of planar rarefaction wave to a 2D hyperbolic-elliptic coupling system of the radiating gas on half-space

This paper studies the asymptotic stability of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupling system on the two-dimensional half-space, where the data on the boundary and at the far field are prescribed as u_{-} and u_{+} , respectively. We show that the solution to the problem converges to the corresponding planar rarefaction wave for 0\le u_{-}&lt;u_{+} as time tends to infinity. To the best of our knowledge, the stability results of planar rarefaction waves on half-space focus primarily on the single viscous conservation law because the rarefaction wave (one-dimensional diffusion wave) of the corresponding one-dimensional problem to scalar viscous conservation law is known. In other words, for a general high-dimensional system of equations, we cannot obtain the stability of planar rarefaction waves on half-space because we cannot construct the rarefaction wave of the corresponding one-dimensional problem. In this paper, we use the structure of the hyperbolic-elliptic coupling system to obtain the monotonic rarefaction wave of the corresponding one-dimensional hyperbolic-elliptic coupling system, and hence give the stability of the planar rarefaction wave on half-space. This can be viewed as the first result for the system of equations on the stability of planar rarefaction waves on half-space.

Investigation of the ultrasound-induced collapse of air bubbles near soft materials

A numerical investigation into the ultrasound-induced collapse of air bubbles near soft materials, utilizing a novel multi-material diffuse interface method (DIM) model with block-structured adaptive mesh refinement is presented. The present work expands from a previous five-equation DIM by incorporating Eulerian hyperelasticity. The model is applicable to any arbitrary number of interacting fluid and solid material. A single conservation law for the elastic stretch tensor enables tracking the deformations for all the solid materials. A series of benchmark cases are conducted, and the solution is found to be in excellent agreement against theoretical data. Subsequently, the ultrasound-induced bubble-tissue flow interactions are examined. The bubble radius was found to play a crucial role in dictating the stresses experienced by the tissue, underscoring its significance in medical applications. The results reveal that soft tissues primarily experience tensile forces during these interactions, suggesting potential tensile-driven injuries that may occur in relevant treatments. Moreover, regions of maximal tensile forces align with tissue elongation areas. It is documented that while early bubble dynamics remain relatively unaffected by changes in shear modulus, at later stages of the penetration processes and the deformation shapes, exhibit notable variations. Lastly, it is demonstrated that decreasing standoff distances enhances the interaction between bubbles and tissue, thereby increasing the stress levels in the tissue, although the behavior of the bubble dynamics remains largely unchanged.

Generalized symmetries and conservation laws of (1 + 1)-dimensional Klein–Gordon equation

Using advantages of nonstandard computational techniques based on the light-cone variables, we explicitly find the algebra of generalized symmetries of the (1 + 1)-dimensional Klein–Gordon equation. This allows us to describe this algebra in terms of the universal enveloping algebra of the essential Lie invariance algebra of the Klein–Gordon equation. Then, we single out variational symmetries of the corresponding Lagrangian and compute the space of local conservation laws of this equation, which turns out to be generated, up to the action of generalized symmetries, by a single first-order conservation law. Moreover, for every conservation law, we find a conserved current of minimal order contained in this conservation law.

Theory of Diffusive Fluctuations.

The recently developed effective field theory of fluctuations around thermal equilibrium is used to compute late-time correlation functions of conserved densities. Specializing to systems with a single conservation law, we find that the diffusive pole is shifted in the presence of nonlinear hydrodynamic self-interactions, and that the density-density Green's function acquires a branch point halfway to the diffusive pole, at frequency ω=-(i/2)Dk^{2}. We discuss the relevance of diffusive fluctuations for strongly correlated transport in condensed matter and cold atomic systems.

Numerical Study of the Injection of Carbon Dioxide in a Homogeneous Porous Media

This work proposes a locally conservative and less restrictive algorithm to solve the problem dealt with in [1], i.e. a two-phase flow in a homogeneous porous medium (water and CO2), with mass absorption between the fluid phases and reaction between the CO2 phase and the rock. The latter is modeled by two non-linear hyperbolic equations that represent the transport of the flowing phases for a given velocity field (equations of saturation and concentration). From the numerical point of view, we use the operator splitting technique to properly treat the time scale of each physical phenomenon and a high-order non-oscillatory central-scheme finite volume method for nonlinear hyperbolic equations proposed by [2] that was extended for a system of equations with source terms to treat the equations that govern the saturation and concentration of phases. In addition, with respect to source terms, the mass flux between fluid phases was handled using the flash methodology, whereas kinetic theory was applied for reproducing the changes in porosity and permeability that are caused by the reaction of CO2 with the rock. The same physical trends observed in [1] were obtained in our numerical results which indicate a good predictive capability. Furthermore, this method avoids the difficulties that arise when adopting small time steps enforced by CFL stability restrictions. Finally, the results obtained show that the applicability of the KT method is beyond just a single nonlinear conservation law with the absence of source terms.

Generalized fermion symmetry, its currents algebra, and Ward-Takahashi identities

We introduce a new local symmetry into the fermion sector of a gauge invariant Lagrangian which may or may not contain a scalar or spontaneous symmetry breaking. The Standard Model in the unitary gauge and QCD are particular cases where this symmetry may apply. We determine the associated vector and axial vector currents and their conservation laws. We show that a single current conservation law may lead to multiple Ward-Takahashi identities. Our results can potentially have important consequences for effective models of low-energy QCD and hadron structure. As an specific example, we discuss the construction of tetraquark states within a generalized linear sigma model and show that this new symmetry probes the tetraquarks in a manner that is consistent with the large $N_c$ limit of QCD.

Derivative artificial compression method for conservation laws with multi-dimensional extension

ABSTRACTA new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.

Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws

This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well-balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.

Dynamic Finite Size Effects in Spiking Neural Networks

We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion. The small parameter for the expansion is the inverse number of neurons in the network. The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas. The Klimontovich equation does not possess well-behaved solutions but can be recast in terms of a coupled system of well-behaved moment equations, known as a moment hierarchy. The moment hierarchy is impossible to solve but in the mean field limit of an infinite number of neurons, it reduces to a single well-behaved conservation law for the mean neuron density. For a large but finite system, the moment hierarchy can be truncated perturbatively with the inverse system size as a small parameter but the resulting set of reduced moment equations that are still very difficult to solve. However, the entire moment hierarchy can also be re-expressed in terms of a functional probability distribution of the neuron density. The moments can then be computed perturbatively using methods from statistical field theory. Here we derive the complete mean field theory and the lowest order second moment corrections for physiologically relevant quantities. Although we focus on finite-size corrections, our method can be used to compute perturbative expansions in any parameter.

On alternative Poisson brackets for fluid dynamical systems and their extension to Stokes-Dirac structures

In this paper we shall consider the Hamiltonian formulation of one-dimensional models of fluid dynamical systems such as the Korteweg de Vries equation and the Boussinesq equation and their extension to port-Hamiltonian systems. We consider the Korteweg de Vries equation and recall its bi-Hamiltonian structure, either formulated as a single conservation law or formulated with respect to Magri's bracket. In both cases we give an extension of the associated Hamiltonian operators to a Stokes-Dirac structure. Then we consider the Boussinesq equation and recall the two Hamiltonian representations either with respect to the canonical Hamiltonian operator associated with a system of two coupled conservation laws or with respect to a third order Hamiltonian operator. In this case we shall suggest a third Hamiltonian operator for which an extension to a Dirac structure is derived.

Time and spatially resolved quench of the fermionic Hubbard model showing restricted equilibration

We investigate the quench of half-filled 1D and 2D fermionic Hubbard models to models without Coulomb interaction. Since the time propagation is gaussian we can use a variety of time-dependent quantum Monte Carlo methods to tackle this problem without generating a dynamical sign problem. Using a continuous time quantum Monte Carlo method (CTQMC) we achieve a system size of 128 sites in 1D, and using a Blankenbecler-Scalapino-Sugar (BSS) type algorithm we were able to simulate 20 x 20 square lattices. Applying these methods to study the dynamics after the quench, we observe that the final state of the system can be reasonably well described by a thermal single-particle density matrix that takes the initial single particle conservation laws into account. The characteristic decay towards this limit is found to be oscillatory with an additional power law decay that depends on the dimensionality. This numerically exact result is shown to compare favorable to mean-field approximations as well as to perturbation theory. Furthermore we observe the information propagation in the 1D-case in the charge charge and spin spin correlations and find that it is linear with a velocity of roughly v = 4 in units of the hopping amplitude.

Dynamical Simulation of Integrable and Nonintegrable Models in the Heisenberg Picture

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient. Using the spin-1/2 XX chain as generic example of an integrable model that can be mapped to free fermions, we provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature autocorrelation function which decays slower than exponential, i.e., with a power law. Thus efficient simulability may already be implied by a single conservation law as we will illustrate numerically for the spin-1 XXZ model.

Criteria for singularities of smooth maps from the plane into the plane and their applications

We give useful criteria for lips, beaks and swallowtail singularities of a smooth map from the plane into the plane. As an application of the criteria, we discuss the singularities for a Cauchy problem of single conservation law.

Semi-Godunov schemes for multiphase flows in porous media

We describe a class of finite volume schemes for 2 × 2 systems of conservations laws based on a “local” decomposition of the system into a series of single conservation laws but with discontinuous coefficients. The resulting schemes are based on Godunov type solvers of the reduced equations. These schemes are very easy to implement since they do not use detailed information about the eigenstructure of the full system. We illustrate the efficiency of the schemes on a variety of numerical experiments focusing on three-phase flows in porous media and show that they are robust and approximate the flow very well, even in the presence of gravity.

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234–2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806–811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.

Third order equivalent equation of lattice Boltzmann scheme

We recall the origin of lattice Boltzmann scheme and detail the version due to D'Humières [8]. We present a formal analysisof this lattice Boltzmann scheme in terms of a single numericalinfinitesimal parameter. We derive third order equivalent partial differential equation of this scheme.Both situations of single conservation law and fluid flow with mass and momentum conservations are detailed. We apply our analysis to so-called D1Q3 and D2Q9 lattice Boltzmann schemes in one and two space dimensions.

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.

Semi-Godunov schemes for general triangular systems of conservation laws

General m × m triangular systems of conservation laws in one space dimension are considered. These systems arise in applications like multi-phase flows in porous media and are non-strictly hyperbolic. Simple and efficient finite-volume schemes of the Godunov type are devised. These are based on a local decoupling of the system into a series of single conservation laws with discontinuous coefficients and are hence termed semi-Godunov schemes. These schemes are not based on the characteristic structure of the system. Some useful properties of the schemes are derived and several numerical experiments demonstrate their robustness and computational efficiency.