Discrete Entropy Research Articles

Convergence of an Implicit Spacetime Godunov Finite Volume Method for a Class of Hyperbolic Systems

We study an implicit spacetime Godunov method for a class of hyperbolic conservation laws known as Temple class systems. We establish the well‐posedness of this method, a discrete entropy inequality, a property analogous to the total variation diminishing property of certain numerical schemes for scalar conservation laws, and, as a consequence, the convergence of the numerical method.

Analysis of a parabolic cross-diffusion population model without self-diffusion

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H 1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler–Galerkin approximation, discrete entropy estimates, and L 1 weak compactness arguments. Furthermore, employing the entropy–entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.

A relaxation scheme for the approximation of the pressureless Euler equations

AbstractIn the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δ‐shocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed first‐order scheme but also a classical MUSCL‐type second‐order extension confirm the reliability and robustness of the relaxation approach. The article extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established, and two‐dimensional numerical results are presented. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

The Implicit Upwind Method for 1-D Scalar Conservation Laws with Continuous Fluxes

We present the first part of a theory of monotone implicit methods for scalar conservation laws. In this paper, we focus on the implicit upwind scheme. The theoretical investigation of this method is centered around a rigorously verified implicit monotonicity criterion. The relation between the upwind scheme and a discrete entropy inequality is constructed analogously to the classical approach of Crandall and Majda [M. G. Crandall and A. Majda, Math. Comp., 34 (1980), pp. 1--21]. A proof of convergence is given which does not rely on a classical compactness argument. The theoretical results are complemented by a discussion of numerical aspects.

Inégalités d'entropie pour un schéma de relaxation

This work is devoted to the discrete entropy inequalities when considering relaxation schemes. After describing the numerical method, we propose a direct proof to establish the discrete entropy inequalities. In fact, we show that the considered relaxation model satisfies a minimum principle on the entropy. This principle implies the expected inequalities. The work is concluded when applying the above results to the 10 moment model. To cite this article: C. Berthon, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Rényi continuous entropy of DNA sequences

Entropy measures of DNA sequences estimate their randomness or, inversely, their repeatability. L-block Shannon discrete entropy accounts for the empirical distribution of all length-L words and has convergence problems for finite sequences. A new entropy measure that extends Shannon's formalism is proposed. Rényi's quadratic entropy calculated with Parzen window density estimation method applied to CGR/USM continuous maps of DNA sequences constitute a novel technique to evaluate sequence global randomness without some of the former method drawbacks. The asymptotic behaviour of this new measure was analytically deduced and the calculation of entropies for several synthetic and experimental biological sequences was performed. The results obtained were compared with the distributions of the null model of randomness obtained by simulation. The biological sequences have shown a different p-value according to the kernel resolution of Parzen's method, which might indicate an unknown level of organization of their patterns. This new technique can be very useful in the study of DNA sequence complexity and provide additional tools for DNA entropy estimation. The main MATLAB applications developed and additional material are available at the webpage http://bioinformatics.musc.edu/renyi. Specialized functions can be obtained from the authors.

Dynamical Shannon entropy and information Tsallis entropy in complex systems

In this work, we offer a new approach to the concept of the information entropy for the description of the dynamic behaviour of complex systems. On the basis of the unification of the known information approach to the Shannon entropy, submitted in works of authors [Phys. Rev. E 62 (5) (2000) 6178; Physica A 303 (2002) 427], and generalization of the Boltzman–Gibbs entropy, offered by Tsallis [Braz. J. Phys. 29 (1) (1999) 1], we have received a new representation of the dynamic information entropy. Here we present concrete applications of the received equations to the study of complex systems related to the electric signals of ECGs of healthy people and patients with myocardial infarction (MI) as an example. Various modifications of the non-Markovity parameter are also submitted. They were received with the help of the new approach to the information entropy. We have received detailed information about Markov, quasi-Markov and non-Markov characteristics of the RR-interval fluctuations of the ECGs with the help of the chain of Zwanzig’-Mori's discrete kinetic equations and dynamical Tsallis entropy.

A note on (2K+1)-point conservative monotone schemes

First-order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a (2K + 1)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.

Sharp CFL, Discrete Kinetic Formulation, and Entropic Schemes for Scalar Conservation Laws

We consider semidiscrete and fully discrete conservative finite volume schemes approximating the solution to one-dimensional scalar conservation law. We show that all E-schemes are associated with a discrete kinetic formulation with a nonnegative kinetic defect measure. This construction provides an alternative proof of the discrete local entropy inequalities with simple expressions of the discrete entropy fluxes. In contrast to the known results, which are restricted to CFL of the form $\lambda Q\leq 1/2$, our proof holds under sharp CFL conditions.

Numerical Approximations of Pressureless and Isothermal Gas Dynamics

We study several schemes of first- or second-order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations. The pressureless gas system is weakly hyperbolic, giving rise to the formation of density concentrations known as delta-shocks. For the isothermal gas system, the infinite speed of expansion into vacuum leads to zero timestep in the Godunov method based on exact Riemann solver. The schemes we consider are able to bypass these difficulties. They are proved to satisfy positiveness of density and discrete entropy inequalities, to capture the delta-shocks, and to treat data with vacuum.

Entropy satisfying flux vector splittings and kinetic BGK models

We establish forward and backward relations between entropy satisfying BGK relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes.We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support.A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers.We deduce the construction of finite velocity relaxation systems for gas dynamics, and obtain a HLLC scheme for which we are able to prove positiveness of density and internal energy, and discrete entropy inequalities.

Data analysis and entropy steered discrete filtering for the numerical treatment of conservation laws

AbstractThis paper describes the construction of a discrete filter algorithm for second‐order oscillatory schemes applied to scalar conservation laws in two space dimensions. Starting from a modification of the Lax–Wendroff scheme proposed by LeVeque (Numer. Meth. Fluids 1993; 4: 175; SIAM J. Numer. Anal. 1996; 33(2): 627) we extend this algorithm with an anisotropic diffusion procedure in order to smooth the spurious oscillations in the vicinity of a shock. To locate the regions identified with the discontinuity we analyse the discrete data with the help of an indicator for entropy production origin from a discrete entropy inequality. Copyright © 2002 John Wiley & Sons, Ltd.

Fully Discrete, Entropy Conservative Schemes of ArbitraryOrder

We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].

Equilibrium schemes for scalar conservation laws with stiff sources

We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

On some upwind difference schemes for the phenomenological sedimentation-consolidation model

In one space dimension, the phenomenological sedimentation-consolidation model reduces to an initial-boundary value problem (IBVP) for a nonlinear strongly degenerate convection-diffusion equation with a non-convex, time-dependent flux function. The frequent assumption that the effective stress of the sediment layer is a function of the local solids concentration only which vanishes below a critical concentration value causes the model to be of mixed hyperbolic-parabolic nature. Consequently, its solutions are discontinuous and entropy solutions must be sought. In this paper, first a (short) guided visit to the mathematical (entropy solution) framework in which the well-posedness of this and a related IBVP can be established is given. This also includes a short discussion of recent existence and uniqueness results for entropy solutions of IBVPs. The entropy solution framework constitutes the point of departure from which numerical methods can be designed and analysed. The main purpose of this paper is to present and demonstrate several finite-difference schemes which can be used to correctly simulate the sedimentation-consolidation model in civil and chemical engineering and in mineral processing applications, i.e., conservative schemes satisfying a discrete entropy principle. Here, finite-difference schemes of upwind type are considered. To some extent, also stability and convergence properties of the proposed schemes are discussed. Performance of the proposed schemes is demonstrated by simulation of two cases of batch settling and one of continuous thickening of flocculated suspensions. The numerical examples focus on a detailed error study, an illustration of the effect of varying the initial datum, and on simulation of practically important thickener operations, respectively.

Convergence of a discrete-velocity model for the Boltzmann-BGK equation

We prove the convergence of a conservative and entropic discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. The main difficulty, due to its implicit definition, is to prove that this approximation is consistent. We also demonstrate the existence and uniqueness of a solution to the discrete-velocity model, by using a fixed-point theorem. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.

Comments on "Resolution in time-frequency"

The author comments that the first line of conjecture 5 in the paper of DeBrunner, Ozaydin and Przebinda (see ibid., vol.47, p.783-88, 1999) was proven by Dembo et al. (1991). The proof is a consequence of the properties of the discrete Renyi entropies. Conjecture 5 is a special case of the Renyi-Hirschman inequality.

Numerical methods for the simulation of the settling of flocculated suspensions

For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic–parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.

Steady state and picosecond time-resolved fluorescence studies on native, desulpho and deflavo xanthine oxidase

Steady state and time-resolved fluorescence studies on native, desulpho and deflavo xanthine oxidase (XO) have been carried out to investigate the conformational changes associated with the replacement of the molybdenum double bonded sulphur by oxygen and the removal of the flavin adenine dinucleotide (FAD). The steady state quenching experiments of the intrinsic tryptophan residues of the enzyme show that all the nine tryptophans are accessible to neutral quencher, acrylamide, in the native as well as desulpho and deflavo enzymes. However, the number of the tryptophan residues accessible to the ionic quenchers, potassium iodide and cesium chloride, increases upon removal of the FAD centre from the enzyme. This indicates that two tryptophan residues move out from the core of the enzyme to the solvent upon the removal of the FAD. The time-resolved fluorescence studies were carried out on the native, desulpho and deflavo XO by means of the time-correlated single photon counting technique, and the data were analysed by discrete exponential and maximum entropy methods. The results show that the fluorescence decay curve fitted best to a three-exponential model with lifetimes τ 1=0.4, τ 2=1.4 and τ 3=3.0 ns for the native and desulpho XO, and τ 1=0.7, τ 2=1.7 and τ 3=4.8 ns for the deflavo XO. The replacement of the molybdenum double bonded sulphur by oxygen in the desulpho enzyme does not cause any significant change of the lifetime components. However, removal of the FAD centre causes a significant change in the shortest and longest lifetime components indicating a conformational change in the deflavo XO possibly in the flavin domain. Decay-associated emission spectra at various emission wavelengths have been used to determine the origin of the lifetimes. The results show that τ 1 and τ 3 of the native and desulpho XO originate from the tryptophan residues which are completely or partially accessible to the solvent but τ 2 corresponds to those residues which are buried in the core of the enzyme and not exposed to the solvent. For deflavo enzyme, τ 2 is red shifted compared to the native enzyme indicating the movement of tryptophan residues from the core of the enzyme to the solvents.

PREDICTIVE ENTROPY BASED CORRECTION OF PHASE CHANGE COMPUTATIONS WITH FLUID FLOW - PART 1: SECOND LAW FORMULATION

A numerical formulation involving the Second Law of Thermodynamics is examined in the analysis of phase change problems with fluid flow. The discretized entropy transport equation and entropy boundary conditions are described for solid-liquid systems. In addition, the downward concavity and compatibility properties of entropy are applied in a discrete entropy based stability analysis. Discrete analogies of the Second Law provide a physical basis for nonlinear phase-temperature iterations in the energy equation. The numerical formulation includes both corrective steps for accuracy improvements (entropy based diffusivity) as well as predictive steps (entropy based time constraint) for stable computations. It is anticipated that this Second Law formulation can provide an effective enhancement for accurate simulations in phase change problems with fluid flow.