In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson--Nernst--Planck (PNP) equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain $S$-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the $S$-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the $S$-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis, transition dynamics, from a steady-state solution on the unstable branch to a one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.

中文翻译：

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具有空间相互作用的 Poisson--Nernst--Planck 方程的多重稳态解的滞后和线性稳定性分析

在这项工作中，我们数值研究了一类具有狄利克雷边界条件的空间泊松-能斯特-普朗克 (PNP) 方程的多个稳态解的线性稳定性，该方程适用于离子通道。通过数值发现的多个稳态解，我们获得了 $S$ 形的电流-电压和电流-浓度曲线，显示了离子电导对电压和具有记忆效应的边界浓度的滞后响应。提出了边界值问题来定位分岔点并预测$S$形曲线上分岔点附近的局部分岔图。开发了线性稳定性分析的数值方法，以了解仅在数值上可用的稳态解的稳定性。提出了有限差分方案来解决涉及微分算子的导出特征值问题。线性稳定性分析表明，$S$形曲线具有两个不同电导水平的线性稳定分支和一个线性不稳定中间分支，表现出经典的双稳态滞后。正如在线性稳定性分析中预测的那样，从不稳定分支上的稳态解到稳定分支上的稳态解的转换动力学是由与主要特征值模式相关的扰动导致的。进一步的数值试验表明，线性稳定性分析中提出的有限差分格式是二阶精确的。