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

   https://doi.org/10.1090/mcom/3642  

   Liu, Chun; Wang, Cheng; Wise, Steven; Yue, Xingye; Zhou, Shenggao (September 2021, Mathematics of Computation)

   null (Ed.)

   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. 

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2. Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

   https://doi.org/10.1093/imanum/drac014  

   Hu, Jingwei; Zhang, Xiangxiong (May 2022, IMA Journal of Numerical Analysis)

   Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions. 

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3. Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential

   https://doi.org/10.1090/mcom/3916  

   Guo, Yunzhuo; Wang, Cheng; Wise, Steven; Zhang, Zhengru (September 2024, Mathematics of Computation)

   A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, is approximated in a semi-implicit manner. In the spatial discretization, the marker and cell difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis. 

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4. Holder estimates for kinetic Fokker-Planck equations up to the boundary

   Luis Silvestre (December 2022, Ars inveniendi analytica)

   We obtain local Holder continuity estimates up to the boundary for a kinetic Fokker-Planck equation with rough coefficients, with the prescribed influx boundary condition. Our result extends some recent developments that incorporate De Giorgi methods to kinetic Fokker-Planck equations. We also obtain higher order asymptotic estimates near the incoming part of the boundary. In particular, when the equation has a zero boundary conditions and no source term, we prove that the solution vanishes at infinite order on the incoming part of the boundary. 

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5. Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

   https://doi.org/10.46298/mna.7203  

   Liu, Jian-Guo; Wang, Ziheng; Xie, Yantong; Zhang, Yuan; Zhou, Zhennan (November 2021, Mathematical Neuroscience and Applications)

   In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the Fokker-Planck equation for the processwith random discharges (i.e. Poisson jumps) are defined on the whole space,while the equation for the limit process is on the half space. However, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomial-order convergence forthe distribution by a re-normalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models. 

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