Convergence Analysis of Structure-Preserving Numerical Methods Based on Slotboom Transformation for the Poisson--Nernst--Planck Equations

Abstract

The analysis of structure-preserving numerical methods for the Poisson-Nernst-Planck (PNP) system has attracted growing interests in recent years. A class of numerical algorithms have been developed based on the Slotboom reformulation, and the mass conservation, ionic concentration positivity, free-energy dissipation have been proved at a discrete level. Nonetheless, a rigorous convergence analysis for these Slotboom reformulation-based, structure-preserving schemes has been an open problem for a long time. In this work, we provide an optimal rate convergence analysis and error estimate for finite difference schemes based on the Slotboom reformulation. Different options of mobility average at the staggered mesh points are considered in the finite-difference spatial discretization, such as the harmonic mean, geometric mean, arithmetic mean, and entropic mean. A semi-implicit temporal discretization is applied, which in turn results in a non-constant coefficient, positive-definite linear system at each time step. A higher order asymptotic expansion is applied in the consistency analysis, and such a higher order consistency estimate is necessary to control the discrete maximum norm of the concentration variables. In convergence estimate, the harmonic mean for the mobility average, which turns out to bring lots of convenience in the theoretical analysis, is taken for simplicity, while other options of mobility average would also lead to the desired error estimate, with more technical details involved. As a result, an optimal rate convergence analysis on concentrations, electric potential, and ionic fluxes is derived, which is the first such results for the structure-preserving numerical schemes based on the Slotboom reformulation. It is remarked that the convergence analysis leads to a theoretical justification of the conditional energy dissipation analysis, which relies on the maximum norm bounds of the concentration and the gradient of the electric potential. Some numerical results are also presented to demonstrate the accuracy and structure-preserving performance of the associated schemes.

Structure-preserving numerical method for Ampere-Nernst-Planck model

“…. Although this assumption can be validated by the convergence analysis based on the higher order asymptotic expansion [3,19], whether they could be applied on the ANP model is still open. Fortunately, in practical simulations, the computed solutions are always bounded, so our theoretical results are still valid in this version.…”

“…The Slotboom variables have also been used to derive a fully implicit scheme with unconditional structure-preserving properties in [15]. The convergence analysis for structurepreserving finite difference methods based on the Slotboom transform has been developed as well [3]. Recently, nonlinear implicit schemes based on the energy gradient flow formulation have also been proposed in the works [19,39].…”

A dynamic mass transport method for Poisson-Nernst-Planck equations

“…However, the PNP system is a strongly coupled system of nonlinear equations, also, the PNP system as a gradient flow can take very long time evolution to reach steady states. Hence, designing efficient and stable numerical methods for the PNP system remains an active area of research (see, e.g., [8,22,25,28,35,36]).…”