Energetic Variational Approach Research Articles

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

A two species micro–macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach

Wormlike micelles are self-assemblies of polymer chains that can break and recombine reversibly. In this paper, we derive a thermodynamically consistent two-species micro–macro model of wormlike micellar solutions by employing an energetic variational approach. The model incorporates a breakage and combination process of polymer chains into the classical micro–macro dumbbell model of polymeric fluids in a unified variational framework. We also study different maximum entropy closure approximations to the micro-macro model by “variation-then-closure” and “closure-then-variation” approaches. By imposing a proper dissipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”, preserves the thermodynamical structure of both mechanical and chemical parts of the original system. Several numerical examples show that the closure model can capture the key rheological features of wormlike micellar solutions in shear flows.

Structure-Preserving Numerical Methods for Nonlinear Fokker--Planck Equations with Nonlocal Interactions by an Energetic Variational Approach

In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with external potentials, optimal transport problems, and aggregation-diffusion models. Based on the energetic variational approach, a trajectory equation is first derived by using the balance between the maximal dissipation principle and the least action principle. By a convex splitting technique, we propose energy dissipating numerical schemes for the trajectory equation. Rigorous numerical analysis reveals that the nonlinear numerical schemes are uniquely solvable, naturally respect mass conservation and positivity at the fully discrete level, and preserve steady states in an admissible convex set, where the discrete Jacobian of flow maps is positive. Under certain assumptions on smoothness and a positive Jacobian, the numerical schemes are shown to be second order accurate in space and first order accurate in time. Extensive numerical simulations are performed to demonstrate several valuable features of the proposed schemes. In addition to the preservation of physical structures, such as positivity, mass conservation, discrete energy dissipation, and steady states, numerical simulations further reveal that our numerical schemes are capable of solving degenerate cases of the Fokker--Planck equations effectively and robustly. It is shown that the developed numerical schemes have convergence order even in degenerate cases with the presence of solutions having compact support and can accurately and robustly compute the waiting time of free boundaries without any oscillation. The limitation of numerical schemes due to a singular Jacobian of the flow map is also discussed.

Field theory of reaction-diffusion: Law of mass action with an energetic variational approach.

We extend the energetic variational approach so it can be applied to a chemical reaction system with general mass action kinetics. Our approach starts with an energy-dissipation law. We show that the chemical equilibrium is determined by the choice of the free energy and the dynamics of the chemical reaction is determined by the choice of the dissipation. This approach enables us to couple chemical reactions with other effects, such as diffusion and drift in an electric field. As an illustration, we apply our approach to a nonequilibrium reaction-diffusion system in a specific but canonical setup. We show by numerical simulations that the input-output relation of such a system depends on the choice of the dissipation.

On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials

AbstractThis paper is concerned with well‐posedness of the Cahn–Hilliard equation subject to a class of new dynamic boundary conditions. The system was recently derived in Liu–Wu (Arch. Ration. Mech. Anal. 233 (2019), 167–247) via an energetic variational approach and it naturally fulfils three physical constraints such as mass conservation, energy dissipation and force balance. The target problem examined in this paper can be viewed as a transmission problem that consists of Cahn–Hilliard type equations both in the bulk and on the boundary. In our approach, we are able to deal with a general class of potentials with double‐well structure, including the physically relevant logarithmic potential and the non‐smooth double‐obstacle potential. Existence, uniqueness and continuous dependence of global weak solutions are established. The proof is based on a novel time‐discretization scheme for the approximation of the continuous problem. Besides, a regularity result is shown with the aim of obtaining a strong solution to the system.

Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

Convergence of the linearized system for a compressible Navier-Stokes-Poission system

In this paper, we are concerned with the limit for solutions of combining viscid and compressible isentropic Navier-Stokes-Poisson system in a bounded domain Ω ⊂ RN , N ≥ 1, with Dirichlet boundary condition. We first derive Navier-Stokes-Poisson system by using Energetic Variational Approach. Then the convergence of the compressible isentropic Navier-Stokes-Poisson system to the linearized system is proven for the global weak solution and for the case of general initial data with satisfying appropriate conditions.

A new interface capturing method for Allen-Cahn type equations based on a flow dynamic approach in Lagrangian coordinates, I. One-dimensional case

We develop a new Lagrangian approach — flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager [27,28]. Its main advantage, comparing with numerical methods in Eulerian coordinates, is that thin interfaces can be effectively captured with few points in the Lagrangian coordinate. We concentrate in the one-dimensional case and construct numerical schemes for the trajectory equation in Lagrangian coordinate that obey the variational structures, and as a consequence, are energy dissipative. Ample numerical results are provided to show that only fewer points are enough to resolve very thin interfaces by using our flow dynamic approach.

Global existence of the non-isothermal Poisson–Nernst–Planck–Fourier system

In this paper, we use the Energetic Variational Approach to derive a non-isothermal electrokinetic model. The charge transport is described through the Poisson–Nernst–Planck equations with variable temperature, and the heat flux satisfies the Fourier's law. This Poisson–Nernst–Planck–Fourier model satisfies both the first law and second law of thermodynamics as well as the Onsager's reciprocal relations, thus it is thermodynamic-consistent. Finally, we prove the global well-posedness for this model under the smallness assumption of the initial data by the energy method.

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on $f \log f$ as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete $W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimal error order.

On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

In this paper, we present a systematic framework to derive a variational Lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach. Such discrete energetic variational approaches are analogous to energetic variational approaches [39,25] in a semidiscrete level, which provide a basis of deriving variational “semi-discrete equations” and can be applied to a large class of partial differential equations with energetic variational structures. The numerical schemes derived by this framework can inherit the variational structure from the continuous energy-dissipation law. As an illustration, we develop two variational Lagrangian schemes for the multidimensional porous medium equations (PME), based on two different energy-dissipation laws. We focus on the numerical scheme based on the energy-dissipation law with 12∫Ω|u|2dx as the dissipation functional. Several numerical experiments demonstrate the accuracy of this scheme as well as its ability in capturing the free boundary and estimating the waiting time for the PME in both 1D and 2D.

Long-time behavior of the Cahn–Hilliard equation with dynamic boundary condition

We study the long-time behavior, within the framework of infinite dimensional dynamical systems, of the Cahn–Hilliard equation endowed with a new class of dynamic boundary conditions. The system under investigation was recently derived by Liu–Wu (Arch Ration Mech Anal 233:167–247, 2019) via an energetic variational approach such that it naturally fulfills physical properties like mass conservation, energy dissipation and force balance. For the system with regular potentials, we prove the existence of exponential attractors, which also yields the existence of a global attractor with finite fractal dimension. For the system with singular potentials, we obtain the existence of a global attractor in a suitable complete metric space.

Optimal critical mass for the two-dimensional Keller–Segel model with rotational flux terms

Our aim is to show that several important systems of partial differential equations arising in mathematical biology, fluid dynamics and electrokinetics can be approached within a single model, namely, a Keller-Segel-type system with rotational flux terms. In particular, we establish sharp conditions on the optimal critical mass for having global existence and finite time blow-up of solutions in two spatial dimensions. Our results imply that the rotated chemotactic response can delay or even avoid the blow-up. The key observation is that for any angle of rotation α∈(-π, π], the resulting PDE system preserves a dissipative energy structure. Inspired by this property, we also provide an alternative derivation of the general system via an energetic variational approach. ©2020 International Press.

A Variational Lagrangian Scheme for a Phase-Field Model: A Discrete Energetic Variational Approach

In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states of the original Allen--Cahn type model. Our discretization is based on a prescribed energy-dissipation law in terms of the flow map. By employing a discrete energetic variational approach, this scheme preserves the variational structure of the continuous energy-dissipation law and is energy stable. Plentiful numerical tests show that, by choosing the initial value properly, our method can compute the desired equilibrium state and capture the thin diffuse interface with a small number of mesh points.

On a novel approach for modeling liquid crystalline flows

In this paper, we derive a new model for the description of liquid crystalline flows. While microscopic Doi type models suffer from the high dimensionality of the underlying product space, the more macroscopic Ericksen--Leslie type models describe only the long time behavior of the flow and are valid only close to equilibrium. By applying an energetic variational approach, we derive a new macroscopic model which shall provide an improved description far from equilibrium. The novelty of our approach lies in the way the energy is minimized. Distinguishing between the velocities of particles and fluid allows us to define the energy dissipation not in terms of chemical potentials but in terms of friction induced by the discrepancies in the considered velocities. We conclude this publication by establishing the existence of weak solutions to the newly derived model.

On generalized diffusion and heat systems on an evolving surface with a boundary

We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive the generalized diffusion and heat systems on the evolving surface. Moreover, we investigate the boundary conditions for the two systems to study their conservation and energy laws. As an application, we make a mathematical model for a diffusion process on an evolving double bubble. Specifically, this paper is devoted to deriving the representation formula for the unit outer co-normal vector to the boundary of a surface.

Numerical complete solution for random genetic drift by energetic variational approach

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

Numerical methods for porous medium equation by an energetic variational approach

We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem difficult to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on flog⁡f as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on 1/(2f) as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existing methods. Due to its linear nature, the second scheme is more efficient.

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

The Cahn--Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn--Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as $t\to+\infty$ and characterize the stability of local energy minimizers for the system.

Three-phase Model of Visco-elastic Incompressible Fluid Flow and its Computational Implementation.

Energetic Variational Approach is used to derive a novel thermodynamically consistent three-phase model of a mixture of Newtonian and visco-elastic fluids. The model which automatically satisfies the energy dissipation law and is Galilean invariant, consists of coupled Navier-Stokes and Cahn-Hilliard equations. Modified General Navier Boundary Condition with fluid elasticity taken into account is also introduced for using the model to study moving contact line problems. Energy stable numerical scheme is developed to solve system of model equations efficiently. Convergence of the numerical scheme is verified by simulating a droplet sliding on an inclined plane under gravity. The model can be applied for studying various biological or biophysical problems. Predictive abilities of the model are demonstrated by simulating deformation of venous blood clots with different visco-elastic properties and experimentally observed internal structures under different biologically relevant shear blood flow conditions.