A positive and energy stable numerical scheme for the Poisson–Nernst–Planck–Cahn–Hilliard equations with steric interactions

Qian, Yiran; Wang, Cheng; Zhou, Shenggao

doi:10.1016/j.jcp.2020.109908

Cited by 39 publications

(20 citation statements)

References 68 publications

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“…This gradient flow structure is similar to that of the Poisson-Nernst-Planck (PNP) system [36,45]. As a consequence, the overall system satisfies the energy-dissipation law [50]:…”

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confidence: 60%

“…The implicit treatment of these nonlinear and singular logarithmic terms are crucial to enforce the positivity of the numerical solution, as well as the energy stability analysis, while it has posed a great challenge in the theoretical justification of the convergence analysis. Also see the related works [8,16,17,18,53] for the Cahn-Hilliard equation with Flory-Huggins energy potential, as well as [36,45] for the Poisson-Nernst-Planck system, [19] for the porous medium equation, [56] for a liquid film droplet model, etc.…”

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confidence: 99%

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Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

Li¹,

Wang²,

Wang³

et al. 2021

Preprint

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We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu et al., J. Comput. Phys., 436, 110253, 2021] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms, which are treated implicitly in the chemical reaction stage. In addition, a combination of a rough error estimate and a refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting results in the convergence estimate of the numerical scheme for the full reaction-diffusion system.

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“…This gradient flow structure is similar to that of the Poisson-Nernst-Planck (PNP) system [36,45]. As a consequence, the overall system satisfies the energy-dissipation law [50]:…”

mentioning

confidence: 60%

mentioning

confidence: 99%

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

Li¹,

Wang²,

Wang³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is crucial to add this term to establish the positivity-preserving property of the numerical solution in the admissible set. Also see the related numerical analysis for the Cahn-Hilliard gradient flow with Flory-Huggins energy potential [9,17,18,19], the Poisson-Nernst-Planck (PNP) system [43,54], etc. Remark 3.2.…”

Section: Reaction-diffusion Systemsmentioning

confidence: 99%

A second-order accurate, operator splitting scheme for reaction-diffusion systems in an energetic variational formulation

Li¹,

Wang²,

Wang³

2021

Preprint

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A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting numerical scheme is proposed and analyzed for the system of reactiondiffusion equations with detailed balance. The scheme is designed based on an energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. At the reaction stage, the reaction trajectory equation is approximated by a second-order Crank-Nicolson type method. The unique solvability, positivity-preserving and energy-stability are established based on a convexity analysis. In the diffusion stage, an exact integrator is applied if the diffusion coefficients are constant, and a Crank-Nicolson type scheme is applied if the diffusion process becomes nonlinear. In either case, both the positivity-preserving property and energy stability could be theoretical established. Moreover, a combination of the numerical algorithms at both stages by the Strang splitting approach leads to a second-order accurate, structure preserving scheme for the original reaction-diffusion system. Numerical experiments are presented, which demonstrate the accuracy of the proposed scheme.

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“…In particular, an implicit treatment of the nonlinear and singular logarithmic term played an essential role in the theoretical analysis, since the convex and the singular nature of the implicit logarithmic part prevents the numerical solution from approaching the singular values of −1 and 1. Similar ideas have been applied to other gradient models singular energy potential, such as the Cahn-Hilliard model with Flory-Huggins-deGennes energy potential [25][26][27]66], the Poisson-Nernst-Planck system [50,57], the reaction-diffusion system with detailed balance [49], a liquid film droplet model [68], et cetera.…”

Section: Introductionmentioning

confidence: 99%

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

Chen¹

2022

CICP

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In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

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A positive and energy stable numerical scheme for the Poisson–Nernst–Planck–Cahn–Hilliard equations with steric interactions

Cited by 39 publications

References 68 publications

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

A second-order accurate, operator splitting scheme for reaction-diffusion systems in an energetic variational formulation

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

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