Conservation Laws Research Articles

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Asymptotic analysis on bright solitons and breather solutions of a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

Abstract We study a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide. We obtain the Lax pair and N-fold Darboux transformation (DT) with N being a positive integer. Based on Lax pair obtained by us, we derive the infinitely-many conservation laws. We give the bright one-, two-, and N-soliton solutions, and the first-, second-, and Nth-order breather solutions based on the N-fold DT. We conclude that the velocities of the bright solitons are influenced by the distributed gain function, g(z), and variable coefficients in equation, h 1(z), p 1(z), r 1(z), and s 1(z) via the asymptotic analysis, where z represents the propagation variable or spatial coordinate. We also graphically observe that: the velocities of the first- and second-order breathers will be affected by h 1(z), p 1(z), r 1(z), and s 1(z), and the background wave depends on g(z).

Uniformly high-order bound-preserving OEDG schemes for two-phase flows

This paper proposes and rigorously analyzes novel high-order bound-preserving oscillation-eliminating discontinuous Galerkin (BP-OEDG) schemes with the Harten–Lax–van Leer (HLL) numerical flux for the Kapila five-equation two-phase flow model. The evolution equation for the volume fraction is formulated as a conservative advection equation with an additional non-conservative source term. Utilizing a technical splitting approach, we design a uniformly high-order discretization for this source term, incorporating an “upwind” discretization of the non-conservative product at cell interfaces based on the HLL wave speeds. This method inherently ensures the symmetry of the volume fractions, which is crucial for establishing the bound-preserving property. The positivity of partial densities and internal energy is rigorously proven using the geometric quasilinearization (GQL) approach, which transforms the nonlinear pressure positivity constraint into equivalent linear constraints. To suppress potential spurious oscillations, we incorporate a scale-invariant and linearity-invariant oscillation elimination (OE) procedure that damps the DG modal coefficients after each Runge–Kutta stage, as proposed in [M. Peng, Z. Sun and K. Wu, OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws, Math. Comput. (2024), https://doi.org/10.1090/mcom/3998]. This OE procedure, acting as a post-processing filter based on the jumps of the DG solution at cell interfaces, is easy to implement, maintains the Abgrall equilibrium condition around an isolated material interface, and preserves the high-order accuracy of the DG schemes. The effectiveness and robustness of the proposed high-order BP-OEDG schemes are demonstrated through several benchmark numerical experiments.

Theory of high-temperature superfluorescence in hybrid perovskite thin films.

The recent discovery of high-temperature superfluorescence in hybrid perovskite thin films has opened new possibilities for harnessing macroscopic quantum phenomena in nanotechnology. This study aimed to elucidate the mechanism that enables high-temperature superfluorescence in these systems. The proposed model describes a quasi-2D Wannier exciton in a thin film that interacts with phonons via the longitudinal optical phonon-exciton Fröhlich interaction. We show that the super-radiant properties of the coherent state in hybrid perovskites are stable against perturbations caused by the longitudinal optical phonon-exciton Fröhlich interaction. Using the multiconfiguration Hartree approach, we derive semiclassical equations of motion for a single-exciton wavefunction, where the vibrational degrees of freedom interact with the Wannier exciton through a mean-field Hartree term. Super-radiance is effectively described by a non-Hermitian term in the Hamiltonian. The analysis was then extended to multiple excited states using the semiclassical Hamiltonian as the basic model. We demonstrate that the ground state of the model exciton Hamiltonian with long-range interactions is a symmetric Dicke super-radiant state, where the Fröhlich interaction is nullified. The additional density matrix-based consideration draws an analogy between this system and stable systems, where the conservation laws determine the nullification of the constant (momentum-independent) decay rate part. In the exciton-phonon system, nullification is associated with the absence of a momentum-independent component in the Wannier exciton-phonon interaction coupling function.

A problem with the conservation law observed in macroscopic quantum phenomena is a consequence of violation of the correspondence principle

Jorge Hirsch drew attention to the contradiction of the Meissner effect with Faraday’s law and the law of angular momentum conservation. This article draws attention to the fact that the angular momentum can change without any force only due to quantization in both microscopic and macroscopic quantum phenomena. But if in the first case this change cannot exceed the Planck constant, then in the second case this change is macroscopic due to a violation of the correspondence principle. To explain the violation of the correspondence principle, Lev Landau postulated in 1941 that microscopic particles in superfluid helium and superconductor cannot move separately. Quantization can change not only the angular momentum, but also the kinetic energy of a superconducting condensate on a macroscopic amount. For this reason, the Meissner effect and other macroscopic quantum phenomena contradict the second law of thermodynamics. The reluctance of physicists to admit the violation of the second law of thermodynamics provoked a false understanding of the phenomenon of superconductivity and obvious contradictions in books on superconductivity.

Design of a long depth of focus Gaussian beam shaping system via aspheric lenses

This paper presents a beam shaping system that uses a Galilean beam shaper. The light mapping function and the aspherical coefficient, which determines the characteristics of the aspheric lens, are derived based on the law of conservation of energy. Furthermore, optical software (i.e., ZEMAX) was utilized to simulate the design of the shaping system. In addition, a spherical collimation system is introduced to achieve the shaping system with long depth of focus. The results show that the output flat-top beam has the uniformity over 90% within the 300mm depth of focus range and the divergence angle no more than 0.6°, which is in line with the theoretical expectation. Compared to the previous works, our beam shaping system is characterized by long depth of focus, simple structure and good beam shaping effect, which makes the present shaping beam more flexible for the application in the field of laser processing.

Simulation of gas mixture dynamics in a pipeline network using explicit staggered-grid discretization

We develop an explicit staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge boundary values that are more complex than for a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

Laws of conservation of mass and momentum for an ideal fluid in the inertial framework of special relativity

In the realm of Newtonian fluids, fundamental conservation laws such as those governing mass, momentum, and energy are rigorously upheld within absolute reference frames. However, within the framework of special relativity and the concept of space -time, the notions of separate conservation of mass and momentum no longer hold. Instead, the theory relies on Lorentz transformations and relative reference systems, typically denoted as S and S', moving uniformly relative to each other, each with its own temporal framework. Within the context of special relativity, particularly in the study of particles, significant findings pertain to the energy-momentum vector. These findings underscore the necessity, when studying fluids within inertial frames, to apply conservation principles to the energy-momentum tensor.

Numerical methods for optimizing logistics systems in the process of inventory management

Scientists have always faced the task of evaluating the effectiveness of the functioning of complex systems, which unambiguously include logistics systems of transport (water, air, rail, automobile and their combinations) and stocks of products (goods), which (stocks of goods) are in static (warehouses) and dynamic (transport) states over time. A huge amount of theoretical material has been accumulated in this field of knowledge, which has been repeatedly tested and successfully applied, especially in relation to logistics systems. However, as is known, on the basis of an efficiency assessment, similar systems are compared, the operation of the systems is promptly evaluated, the system's compliance with its purpose is checked, the presence or absence of targeted targeting is found out, etc. Hence, the relevance of the mathematical apparatus (numerical methods) for optimizing stocks of products (goods) is determined as a calculation for reasoned decision-making on effective inventory management using transport and storage capacities on an economic basis: the objective function is to minimize costs. Cost reduction in the processes of functioning of logistics systems due to the action of two basic laws – the law of saving time and the law of conservation of energy is one of the main modern directions for increasing their efficiency as complex systems [1-5]. Therefore, the optimization of logistics systems according to the criterion of costs (losses) in the processes of their functioning is defined as the formulation of the research task. The aim of the work is to develop the application of numerical methods for optimizing logistics systems in the process of inventory management. The novelty consists in the application of numerical methods and the calculated ratios obtained, which can be used in the practice of the logistics system of water, air, rail and road transport for inventory management by optimizing the determined losses (costs). The result consists in the developed method of studying theoretically calculated losses occurring in the processes of functioning of logistics systems of all types of transport and their combinations (hereinafter transport). The theoretical significance of the results obtained is the development of the application of numerical methods for optimizing logistics systems in the process of inventory management, including: a set of factors and criteria for optimizing logistics processes; a method for modeling and optimizing the processes of functioning of logistics transport systems; a method for analyzing losses in the processes of functioning of logistics transport systems. These theoretical provisions make it possible to optimize the processes of functioning of logistics transport systems and, ultimately, to increase the efficiency of providing consumers, which is of important socio-economic importance.

Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of [1] has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown in [1] that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws, such as the 2×2 system of the shallow water equations and the standard 3×3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation [2]. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method (PPS), with the improved estimate obtained using the novel spectral relaxation scheme.

Photocatalytic antibiotic degradation in coated open microchannels by applying 2D and 3D flow modeling with kinetics

The accumulation of active pharmaceutical ingredients in the aqueous environment is a serious problem that will become even more concerning in the future. In this work, the photocatalytic degradation of ciprofloxacin (CIP) in aqueous solution was assessed over P25-TiO2 coated open microchannels with gravity-driven flow under UV-A irradiation. The deposition of different amounts of TiO2 in the microchannels was carried out via a facile, self-developed procedure. The degradation kinetics of ciprofloxacin was described via the Langmuir-Hinshelwood mechanism. Since the flow characteristics in the microchannel had influence on the concentration distribution of CIP in the microchannel, the coupled momentum and mass conservation law was solved numerically in MATLAB 2023a (2D case) as well as in ANYS Fluent 2023 R1 (2D and 3D cases). Although the implemented 2D model in MATLAB 2023a allowed the preliminary estimation of the selected kinetic parameters, namely adsorption equilibrium constant and specific Langmuir-Hinshelwood rate constant, the sensitivity of the model was not satisfactory which was attributed to the empirical correlations used for the estimation of the external mass transfer coefficient. The 2D and 3D models in ANSYS Fluent 2023 R1 predicted efficiently the outlet concentration of ciprofloxacin for different inlet CIP concentrations and liquid phase flow rates. Therefore, the as developed 2D and 3D models in ANSYS Fluent 2023 R1 can be used for the design of reactors containing coated microchannels with gravity-driven flow for photocatalytic degradation of active pharmaceutical ingredients.

A robust compact finite difference framework for simulations of compressible turbulent flows

A robust high-order compact finite difference framework is proposed for simulations of compressible turbulent flows with high spectral resolution using a fully collocated variable storage paradigm. Both inviscid and viscous fluxes are assembled at the edge-staggered grid locations. Nonlinear robustness is attained as a consequence of the intrinsic reduction of aliasing errors in the inviscid fluxes due to the spectral behavior of the compact interpolation schemes. Additional robustness is provided by enhancing the spectral resolution of the viscous flux and its divergence at small scales using purely staggered numerical differentiation. Demonstrative simulations have shown numerical stability of the compact finite difference discretization without any type of solution filtering on both Cartesian and curvilinear meshes. For simulations on a curvilinear mesh, a general metric evaluation approach that satisfies the geometric conservation law is proposed. Additional approaches to combining the proposed scheme with approximate Riemann solvers and artificial diffusivities for shock-capturing are also discussed. Along with theoretical analysis, rigorous evaluation and validation of the methodology on canonical tests, including classic two-dimensional simulations, direct numerical simulations, and large-eddy simulations, are used to confirm robustness and accuracy.

Conservation laws of a generalized model for propagation pulses with four power nonlinearities

A generalized model for propagation pulses with four power nonlinearities is considered. The equation studied is the generalization of some well-known models and allows us to evaluate the influence of various processes on pulse propagation. The three conservation laws of the equation are found. The equation does pass not the Painlevé test and the Cauchy problem cannot be solved by the inverse scattering transform. Analytical solutions of the generalized nonlinear Schrödinger equation are found taking into account the traveling wave reduction. Optical solitons corresponding to the mathematical model are given. Conservative quantities for the bright optical soliton are calculated.

On thermodynamic processes, state equations and critical phenomena for homogeneous mixtures

In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium.

Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024

Quantum Liquids: Emergent Higher-Rank Gauge Theory and Fractons

Fractons emerge from many-body systems, featuring subdimensional particles with restricted mobility. These particles have attracted interest for their roles across disciplines, including topological quantum codes, quantum field theory, emergent gravity, and quantum information. They display unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review offers a structured overview of fracton phenomena, especially those of gapless fracton liquids, which enable collective modes similar to gauge fluctuations in Maxwell's electromagnetic framework, yet their phenomena are distinguished by a unique conservation law that restricts the mobility of individual charges and monopoles. We delve into the theoretical basis of three-dimensional (3D) fracton liquids, exploring emergent symmetric tensor gauge theories and their properties. We also discuss the material realization of fracton liquids in Yb-based pyrochlore lattices and other synthetic quantum matter platforms.

Conservation law of harmonic mappings in supercritical dimensions

Conservation law of harmonic mappings in supercritical dimensions

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.

ABOUT FREED AND STRUCTURED INFORMATION

The article considers issues related to the study of information transmission in communication channels and the dissemination of their results to the surrounding nature. Two types of information, free and structural, are considered. Their definitions are given and the relationship between them is shown. The question of the purpose of both types of information when transmitting messages is considered. Algorithms for calculating free and structural information are given. The significance of structural information for the theory of vitalism and nature in general is shown. The law of conservation of information in transmitted messages and natural objects is formulated.