Relaxation Operators Research Articles

Numerical and Theoretical Analysis of Model Equations for Multicomponent Rarefied Gas

Model equations approximating the system of Boltzmann equations for a multicomponent gas are investigated. Methods for determining parameters in relaxation terms corresponding to cross-collision integrals are analyzed. Numerical solutions based on three model systems and the Boltzmann equations are compared as applied to the following problems: relaxation of a mixture to equilibrium, shock wave structure, and the dynamics of a vapor-gas cloud generated by pulsed laser irradiation of a target. It is shown that the parameters in the relaxation operators influence the degree of difference in the solutions produced by the various models.

BGK Model for Two-Component Gases Near a Global Maxwellian

In this paper, we establish the existence of the unique global-in-time classical solutions to the two-component Bhatnagar–Gross–Krook (BGK) model suggested in [C. Klingenberg, M. Pirner, and G. Puppo, Kinet. Relat. Models, 10 (2017), pp. 445–465] when the initial data is a small perturbation of global equilibrium. For this, we carefully analyze the dissipative nature of the linearized two-component relaxation operator and observe that the partial dissipation from the intraspecies and the interspecies linearized relaxation operators are combined in a complementary manner to give rise to the desired dissipation estimate of the model. We also observe that the convergence rate of the distribution function increases as the momentum-energy interchange rate between the different components of the gas increases.

Robust multigrid methods for nearly incompressible elasticity using macro elements

Abstract We present a mesh-independent and parameter-robust multigrid solver for the Scott–Vogelius discretisation of the nearly incompressible linear elasticity equations on meshes with a macro element structure. The discretisation achieves exact representation of the limiting divergence constraint at moderate polynomial degree. Both the relaxation and multigrid transfer operators exploit the macro structure for robustness and efficiency. For the relaxation we use the existence of local Fortin operators on each macro cell to construct a local space decomposition with parameter-robust convergence. For the transfer we construct a robust prolongation operator by performing small local solves over each coarse macro cell. The necessity of both components of the algorithm is confirmed by numerical experiments.

Macroscopic equations for inert gas mixtures in different hydrodynamic regimes

Starting from a BGK model for gas mixtures involving sums of relaxation operators, we formally derive Euler and Navier–Stokes equations in different regimes, in the asymptotic limit for proper Knudsen number, with explicit computation of the transport coefficients of viscosity and thermal conductivity. First, we consider a regime dominated by the whole collision phenomena; then, we focus on the case of ɛ−mixtures of heavy and light species, assuming accordingly that the collisions within each component constitute the dominant process. In this latter case we show that the two-scale collision regime leads to a multi-velocity and multi-temperature hydrodynamic description, which emphasizes the distinctive features of each constituent.

Relaxation operator for quasiparticles in a solid

Popular models of the phenomenological relaxation operators that are widely used in the master equation formalism for open condensed-matter systems have significant flaws ranging from limited applicability to violation of fundamental physical principles. We propose a relatively simple universal model of the relaxation operator which is free from these flaws, has a correct static limit, correct direct-current limit in a uniform electric field, includes both interband and intraband transitions, and is valid for an arbitrary dispersion of quasiparticles in a solid. We use the proposed operator to generalize the Lindhard formula and derive explicit expressions for the relaxation operator for Dirac materials with an unconventional energy spectrum of quasiparticles, such as graphene and Weyl semimetals. We compare the linear susceptibility spectra for graphene obtained with different relaxation models and show that the proposed relaxation operator leads to physically meaningful behavior of the susceptibility at low frequencies whereas the existing models become completely invalid.

An efficient, high-order method for solving Poisson equation for immersed boundaries: Combination of compact difference and multiscale multigrid methods

A new efficient and high-order accurate sharp-interface method for solving the Poisson equation on irregular domains and non-uniform meshes is presented. The approach is based on a combination of a fourth-order compact finite difference scheme and a multiscale multigrid (MSMG) method. The key aspect of the new method is that the regular compact finite difference stencil is modified at the irregular grid points near the immersed boundary to obtain a sharp interface solution while maintaining the formal fourth-order accuracy. The MSMG method is designed based on the standard multigrid V-cycle technique to solve the system of equations derived from the fourth-order compact discretization, while the corresponding multigrid relaxation, restriction and prolongation operators are properly constructed for non-uniform grids with immersed boundaries. The contribution of the present work is the design of a fourth-order-accurate Poisson solver whose accuracy, efficiency and computational cost are independent of the complexity of the geometry and the presence or not of an immersed boundary. The new method is demonstrated and validated for a number of problems including smooth and jagged boundaries. The test cases confirm that the new method is fourth-order accurate in the maximum norm whether an immersed boundary is present or not and on uniform or non-uniform meshes. Furthermore, the computational efficiency of the new method is demonstrated with regard to convergence rate and run time, which shows that the MSMG method is equally efficient for domains with immersed boundaries as for simple domains. The new compact difference method is evaluated by comparison with the standard fourth-order (non-compact) finite difference approximation in terms of both accuracy and computational efficiency. The new compact difference scheme yields indeed more accurate numerical solutions. The striking difference between the two schemes is the much higher computational efficiency: The number of V-cycles needed to reach the discretization error is significantly lower for the new compact method compared to the standard difference scheme. As a result, the new compact scheme requires only a fraction of the computer time for convergence in comparison to the standard fourth-order difference scheme.

Comprehensive solutions to the Bloch equations and dynamical models for open two-level systems

The Bloch equation and its variants constitute the fundamental dynamical model for arbitrary two-level systems. Many important processes, including those in more complicated systems, can be modeled and understood through the two-level approximation. It is therefore of widespread relevance, especially as it relates to understanding dissipative processes in current cutting-edge applications of quantum mechanics. Although the Bloch equation has been the subject of considerable analysis in the 70 years since its inception, there is still, perhaps surprisingly, significant work that can be done. This paper extends the scope of previous analyses. It provides a framework for more fully understanding the dynamics of dissipative two-level systems. A solution is derived that is compact, tractable, and completely general, in contrast to previous results. Any solution of the Bloch equation depends on three roots of a cubic polynomial that are crucial to the time dependence of the system. The roots are typically only sketched out qualitatively, with no indication of their dependence on the physical parameters of the problem. Degenerate roots, which modify the solutions, have been ignored altogether. Here the roots are obtained explicitly in terms of a single real-valued root that is expressed as a simple function of the system parameters. For the conventional Bloch equation, a simple graphical representation of this root is presented that makes evident the explicit time dependence of the system for each point in the parameter space. Several intuitive, visual models of system dynamics are developed. A Euclidean coordinate system is identified in which any generalized Bloch equation is separable, i.e., the sum of commuting rotation and relaxation operators. The time evolution in this frame is simply a rotation followed by relaxation at modified rates that play a role similar to the standard longitudinal and transverse rates. These rates are functions of the applied field, which provides information towards control of the dissipative process. The Bloch equation also describes a system of three coupled harmonic oscillators, providing additional perspective on dissipative systems.

Belief revision, minimal change and relaxation: A general framework based on satisfaction systems, and applications to description logics

Belief revision of knowledge bases represented by a set of sentences in a given logic has been extensively studied but for specific logics, mainly propositional, and also recently Horn and description logics. Here, we propose to generalize this operation from a model-theoretic point of view, by defining revision in the abstract model theory of satisfaction systems. In this framework, we generalize to any satisfaction system the characterization of the AGM postulates given by Katsuno and Mendelzon for propositional logic in terms of minimal change among interpretations. In this generalization, the constraint on syntax independence is partially relaxed. Moreover, we study how to define revision, satisfying these weakened AGM postulates, from relaxation notions that have been first introduced in description logics to define dissimilarity measures between concepts, and the consequence of which is to relax the set of models of the old belief until it becomes consistent with the new pieces of knowledge. We show how the proposed general framework can be instantiated in different logics such as propositional, first-order, description and Horn logics. In particular for description logics, we introduce several concrete relaxation operators tailored for the description logic ALC and its fragments EL and ELU, discuss their properties and provide some illustrative examples.

A Review of Time Relaxation Methods

The time relaxation model has proven to be effective in regularization of Navier–Stokes Equations. This article reviews several published works discussing the development and implementations of time relaxation and time relaxation models (TRMs), and how such techniques are used to improve the accuracy and stability of fluid flow problems with higher Reynolds numbers. Several analyses and computational settings of TRMs are surveyed, along with parameter sensitivity studies and hybrid implementations of time relaxation operators with different regularization techniques.

Theoretical Convergence Rate Lower Bounds for Variants of Coarse Mesh Finite Difference to Accelerate Neutron Transport Calculations

The lower bounds for the theoretical convergence rate of variants of the Coarse Mesh Finite Difference (CMFD) method for neutron transport acceleration are studied in this paper by generalization of the method into three categories: artificially diffusive CMFD, flux relaxation, and higher-order spatial prolongation operators. A Fourier analysis of the methods demonstrates that artificial diffusion and flux relaxation are mathematically equivalent and arbitrarily scale the coarse mesh to fine mesh projection (CMP) vector. The high-order spatial prolongation method is shown to affect the shape of the CMP vector. As a result, any of the CMFD variants based on these three sets of modifications correspond to a specific CMP vector. The optimization process is performed for the multidimensional vector, and the minimum spectral radius among all possible CMP vectors is shown to be the theoretical lower bound for the CMFD convergence rate. The spectral radius associated with the CMFD convergence rate lower bound is found to be slightly smaller (less than 0.04) than optimally diffusive CMFD (odCMFD), and the difference between odCMFD to the CMFD lower bound is much smaller than the difference between both standard CMFD and partial current–based CMFD to the CMFD lower bound. In addition, the odCMFD method has a distinct advantage in ease of implementation and minimal overhead. Conversely, the implementation necessary to achieve the CMFD lower bound would be very complicated, especially for two- and three-dimensional problems.

Flexible query processing for SPARQL

Flexible querying techniques can enhance users’ access to complex, heterogeneous datasets in settings such as Linked Data, where the user may not always know how a query should be formulated in order to retrieve the desired answers. This paper presents query processing algorithms for a fragment of SPARQL 1.1 incorporating regular path queries (property path queries), extended with query approximation and relaxation operators. Our flexible query processing approach is based on query rewriting and returns answers incrementally according to their “distance” from the exact form of the query. We formally show the soundness, completeness and termination properties of our query rewriting algorithm. We also present empirical results that show promising query processing performance for the extended language.

Resolution-of-identity stochastic time-dependent configuration interaction for dissipative electron dynamics in strong fields.

In this contribution, we introduce a method for simulating dissipative, ultrafast many-electron dynamics in intense laser fields. The method is based on the norm-conserving stochastic unraveling of the dissipative Liouville-von Neumann equation in its Lindblad form. The N-electron wave functions sampling the density matrix are represented in the basis of singly excited configuration state functions. The interaction with an external laser field is treated variationally and the response of the electronic density is included to all orders in this basis. The coupling to an external environment is included via relaxation operators inducing transition between the configuration state functions. Single electron ionization is represented by irreversible transition operators from the ionizing states to an auxiliary continuum state. The method finds its efficiency in the representation of the operators in the interaction picture, where the resolution-of-identity is used to reduce the size of the Hamiltonian eigenstate basis. The zeroth-order eigenstates can be obtained either at the configuration interaction singles level or from a time-dependent density functional theory reference calculation. The latter offers an alternative to explicitly time-dependent density functional theory which has the advantage of remaining strictly valid for strong field excitations while improving the description of the correlation as compared to configuration interaction singles. The method is tested on a well-characterized toy system, the excitation of the low-lying charge transfer state in LiCN.

Handling failing RDF queries: from diagnosis to relaxation

Recent years have witnessed the development of large knowledge bases (KBs). Due to the lack of information about the content and schema semantics of KBs, users are often not able to correctly formulate KB queries that return the intended result. In this paper, we consider the problem of failing RDF queries, i.e., queries that return an empty set of answers. Query relaxation is one cooperative technique proposed to solve this problem. In the context of RDF data, several works proposed query relaxation operators and ranking models for relaxed queries. But none of them tried to find the causes of an RDF query failure given by Minimal Failing Subqueries (MFSs) as well as successful queries that have a maximal number of triple patterns named Ma$$\underline{x}$$x?imal Succeeding Subqueries (XSSs). Inspired by previous work in the context of relational databases and recommender systems, we propose two complementary approaches to fill this gap. The lattice-based approach (LBA) leverages the theoretical properties of MFSs and XSSs to efficiently explore the subquery lattice of the failing query. The matrix-based approach computes a matrix that records alternative answers to the failing query with the triple patterns they satisfy. The skyline of this matrix directly gives the XSSs of the failing query. This matrix can also be used as an index to improve the performance of LBA. The practical interest of these two approaches are shown via a set of experiments conducted on the LUBM benchmark and a comparative study with baseline and related work algorithms.

СПЕКТРАЛЬНЫЕ РАЗЛОЖЕНИЯ В ДИНАМИЧЕСКИХ ЗАДАЧАХ ВЯЗКОУПРУГОСТИ

Theoretical relations obtained from solutions of dynamic problems of viscoelasticity represent an effective framework for experimental identification of dynamic rheological properties of materials. For the construction of such relations, closed solutions of boundary value problems (i.e. written in the form of convergent series or integrals) are preferred, because they(unlike solutions obtained by numerical methods) allow strict error estimates. However, the construction of analytical solutions is associated with the following difficulties. 1. As usual, the hypothesis of proportionality is accepted for relaxation operators corresponding to the first and second Lame moduli, which is equivalent to the hypothesis of a constant Poisson’s ratio. This significantly reduces the generality of consideration. 2. Representation of solutions for three-dimensional problems in the form of expansions in eigenfunctions makes it necessary to taking into account the large eigenvalues which in the vast majority of problems can be found only numerically, as the roots of transcendental equations, thus, it is likely to skip closely spaced and multiple roots. 3. Constructed series converge slowly. In this paper we suggest ways to overcome these difficulties. Solutions of initial boundary value problems are presented in the form of spectral expansions, but in contrast to the classical method of Fourier decomposition they are expanded over biorthogonal system of eigenfunctions of mutually conjugate pencils of differential operators. This pencils define generalized Sturm-Liouville problem with a polynomial spectral parameter. This eliminates the hypothesis of proportionality relaxation operators. Effective relations for the terms of spectral(in particular normalization factors) coordinate functions and asymptotic formulas for the initial approximations of eigenvalues excluding their omission in calculations are obtained. Power related ranking of elements of the spectral decomposition is proposed which allows achievingthe required accuracy of calculations on the partial sums of a low order.

Reducing Parallel Communication in Algebraic Multigrid through Sparsification

Algebraic multigrid (AMG) is an $\mathcal{O}(n)$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids which utilize complementary relaxation and interpolation operators is constructed. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grid matrices and reduced there. However, large parallel communication costs often limit parallel scalability. As the multigrid hierarchy is formed, each coarse matrix is formed through a triple matrix product. The resulting coarse grids often have significantly more nonzeros per row than the original fine-grid operator, thereby generating high parallel communication costs associated with sparse matrix-vector multiplication (SpMV) on coarse levels. In this paper, we introduce a method that systematically removes entries in coarse-grid matrices after the hierarchy is formed, leading to improved communication costs. We sparsify by removing weakly connected or unimportant entries in the matrix, le...

Thermo-viscoelastic materials with fractional relaxation operators

The new model of linear thermo-viscoelasticity for isotropic media taking into consideration the rheological properties of the volume with fractional relaxation operators is given. The governing equations are taken in a unified system from which some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained. The resulting formulation is applied to several concrete problems, a thermal shock problem and a problem for a half-space subjected to ramp-type heating as well as a problem of a layer media. Laplace transform techniques are used. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effect of fractional relaxation operators and ramping parameter of heating with different theories of thermoelasticity.

Relaxation dynamics in quantum dissipative systems: the microscopic effect of intramolecular vibrational energy redistribution.

We investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.

About the general criteria for correct phenomenological relaxation operators of quantum systems

We define three general conditions that the relaxation operator in the von Neumann equation for the density matrix of an open quantum system should satisfy. The first one provides the correct relation between the macroscopic current and polarization in a quantum medium, the second is defined by thermodynamic arguments and the third one is imposed by gauge invariance of the solution. The corrected phenomenological relaxation operator is proposed for any quantum system with a discrete number of basis functions, as well as for a quantum system described by the density matrix in coordinate representation. The dissipative three-dimensional oscillator in an arbitrary oriented magnetic field is considered in detail. We have shown that the minimally modified relaxation operator in comparison with the standard relaxation operator, corresponding to tau-approximation, eventually leads to simpler and noncontradictory equations for the dynamics of averaged values of physical quantities.

Polaronic quantum master equation theory of inelastic and coherent resonance energy transfer for soft systems

This work extends the theory of coherent resonance energy transfer [S. Jang, J. Chem. Phys. 131, 164101 (2009)] by including quantum mechanical inelastic effects due to modulation of donor-acceptor electronic coupling. Within the approach of the second order time local quantum master equation (QME) in the polaron picture and under the assumption that the bath degrees of freedom modulating the electronic coupling are independent of other modes, a general time evolution equation for the reduced system density operator is derived. Detailed expressions for the relaxation operators and inhomogeneous terms of the QME are then derived for three specific models of modulation in distance, axial angle, and dihedral angle, which are all approximated by harmonic oscillators. Numerical tests are conducted for a set of model parameters. Model calculation shows that the torsional modulation can make significant contribution to the relaxation and dephasing mechanisms.

Exponential Runge–Kutta Methods for Stiff Kinetic Equations

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of BGK type. For Boltzmann-type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.