Discrete Free Energy Research Articles

A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations

A system of two partial differential equations with fractional diffusion is considered in this study. The system extends the conventional Zakharov system with unknowns being nonlinearly coupled complex- and real-valued functions. The diffusion is understood in the Riesz sense, and suitable initial-boundary conditions are imposed on an open and bounded domain of the real numbers. It is shown that the mass and Higgs’ free energy of the system are conserved. Moreover, the total energy is proven to be dissipated, and that both the free and the total energy are non-negative. As a corollary from the conservation of energy, we find that the solutions of the system are bounded throughout time. Motivated by these properties on the solutions of the system, we propose a numerical model to approximate the fractional Zakharov system via finite-difference approaches. Along with this numerical model for solving the continuous system, discrete analogues for the mass, the Higgs’ free energy and the total energy are we provided. Furthermore, utilizing Browder’s fixed-point theorem, we establish the solubility of the discrete model. It is shown that the discrete total mass and the discrete free energy are conserved, in agreement with the continuous case. The discrete energy functionals (both the discrete free energy and the discrete total energy) are proven to be non-negative functions of the discrete time thoroughly the boundedness of the numerical solutions. Properties of consistency, stability and convergence of the scheme are also studied rigorously. Numerical simulations illustrate some of the anticipated theoretical features of our finite-difference solution procedure.

On a discrete model that dissipates the free energy of a time-space fractional generalized nonlinear parabolic equation

The present work is the first manuscript of the literature in which a numerical model that preserves the dissipation of free energy is proposed to solve a time-space fractional generalized nonlinear parabolic system. More precisely, we investigate an extension of the multidimensional heat equation with nonlinear reaction and fractional derivatives in space and time. The model considers homogeneous Dirichlet boundary conditions and initial data. The temporal fractional derivative is understood in the Caputo sense, and Riesz fractional derivatives are employed in the spatial variables. The system possesses a free energy functional and we prove mathematically that it is a decreasing function of time. Motivated by these facts, we propose a discretization of the space-time-fractional model which dissipates the discrete free energy. The discrete model is obtained using fractional-order centered differences to approximate the Riesz derivatives, and the L1 scheme to estimate the Caputo derivatives. The numerical model is a consistent approximation of the continuous system with quadratic order in space, and at most second order in time. Simulations confirm the capability of the numerical model to dissipate the free energy of the continuous fractional system.

A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion

We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to study several one- and two-dimensional systems, with a general free-energy functional accounting for external fields and inter-particle potentials, and placed in non-homogeneous thermal baths characterized by anisotropic, space-dependent and time-dependent properties.

Positive and free energy satisfying schemes for diffusion with interaction potentials

In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, a local scaling limiter is introduced to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.

Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations with General Free Energy

Well-balanced and free energy dissipative first- and second-order accurate finite-volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Lyapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analogous property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories.

A free–energy stable nodal discontinuous Galerkin approximation with summation–by–parts property for the Cahn–Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn–Hilliard equation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The latter permits us to show that the discrete free–energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes. We use the Bassi–Rebay 1 (BR1) scheme to compute interface fluxes, and a first order IMplicit–EXplicit (IMEX) scheme to integrate in time. We provide a semi–discrete stability study, and a fully–discrete proof subject to a positivity condition on the solution. Lastly, we test the theoretical findings using numerical cases that include two and three–dimensional problems.

Entropy Dissipation Semi-Discretization Schemes for Fokker–Planck Equations

We propose a new semi-discretization scheme to approximate nonlinear Fokker–Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric in the space of probability densities. We discretize the underlying state by a finite graph and define a discrete 2-Wasserstein metric in the discrete probability space. Based on such metric, we introduce a gradient flow of the discrete free energy as semi discretization scheme. We prove that the scheme maintains dissipativity of the free energy and converges to a discrete Gibbs measure at exponential dissipation rate. We exhibit these properties on several numerical examples.

A modified Poisson–Nernst–Planck model with excluded volume effect: theory and numerical implementation

The Poisson--Nernst--Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson--Boltzmann (MPB) equation is obtained with the help of the Lambert-W special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.

A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson–Nernst–Planck systems

We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson–Nernst–Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivity of numerical solutions is enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.

Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller---Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in $$L^p$$Lp norm with a suitable $$p>2$$p>2 for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.

A finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors

An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the elec-tric potential. The equations are solved in a bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The main features of the numerical scheme are the preservation of positivity and L ∞ bounds and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is uncon-ditionally stable. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.

A free energy satisfying finite difference method for Poisson–Nernst–Planck equations

In this work we design and analyze a free energy satisfying finite difference method for solving Poisson–Nernst–Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.

Analysis and simulation for an isotropic phase-field model describing grain growth

A phase-field system of coupled Allen--Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well as an exponential attractor, which entails that the global attractor has finite fractal dimension.Moreover, we show that each trajectory converges to a single equilibrium. A time-adaptive numerical scheme based on trigonometric interpolation is presented.It allows to track the approximated long-time behavior accurately and leads to a convergence rate. The scheme exhibits a physically consistent discrete free energy dissipation.

Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction‐diffusion systems

AbstractOur focus are energy estimates for discretized reaction‐diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation.For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The fundamental idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete Sobolev‐Poincaré inequality.

Early Events, Kinetic Intermediates and the Mechanism of Protein Folding in Cytochrome c

Kinetic studies of the early events in cytochrome c folding are reviewed with a focus on the evidence for folding intermediates on the submillisecond timescale. Evidence from time-resolved absorption, circular dichroism, magnetic circular dichroism, fluorescence energy and electron transfer, small-angle X-ray scattering and amide hydrogen exchange studies on the t ≤ 1 ms timescale reveals a picture of cytochrome c folding that starts with the ~ 1-μs conformational diffusion dynamics of the unfolded chains. A fractional population of the unfolded chains collapses on the 1 – 100 μs timescale to a compact intermediate IC containing some native-like secondary structure. Although the existence and nature of IC as a discrete folding intermediate remains controversial, there is extensive high time-resolution kinetic evidence for the rapid formation of IC as a true intermediate, i.e., a metastable state separated from the unfolded state by a discrete free energy barrier. Final folding to the native state takes place on millisecond and longer timescales, depending on the presence of kinetic traps such as heme misligation and proline mis-isomerization. The high folding rates observed in equilibrium molten globule models suggest that IC may be a productive folding intermediate. Whether it is an obligatory step on the pathway to the high free energy barrier associated with millisecond timescale folding to the native state, however, remains to be determined.

Convergence analysis for a smeared crack approach in brittle fracture

Our analysis focuses on the mechanical energies involved in the propagation of fractures: the elastic energy, stored in the bulk, and the fracture energy, concentrated in the crack. We consider a finite element model based on a smeared crack approach: the fracture is approximated geometrically by a stripe of elements and mechanically by a softening constitutive law. We define in this way a discrete free energy $Gh$ (being $h$ the element size) which accounts both for elastic displacements and fractures. Our main interest is the behaviour of $G_h$ as $h \\to 0$. We prove that, only for a suitable choice of the (mesh dependent) constitutive law, $G_h$ converges to a limit functional $G\\phi$ with a positive (anisotropic) term concentrated on the crack. We discuss the mesh bias and compute it explicitly in the case of a structured triangulation.

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.

Multisoliton Behavior of a Ferrielectric Phase in an External Electric Field: Genetic Algorithm Study

A numerical modeling of the three-layer SmC* FI phase behavior under an external electric field is performed. A discrete free energy of the model is analyzed by means of genetic algorithms. We focus our attention on the structure evolution induced by the field and its relation to the field dependence of the in-plane optical anisotropy. We show the multisoliton character of the azimuthal molecular reorientation under the field. It permits us to explain the unusual field-induced behavior of the conoscopic figures in the SmC* FI phase.