An energy-preserving discretization for the Poisson–Nernst–Planck equations

Abstract

The Poisson–Nernst–Planck (PNP) equations have recently been used to describe the dynamics of ion transport through biological ion channels besides being widely employed in semiconductor industry. This paper is about the design of a numerical scheme to solve the PNP equations that preserves exactly (up to roundoff error) a discretized form of the energy dynamics of the system. The proposed finite difference scheme is of second-order accurate in both space and time. Comparisons are made between this energy dynamics-preserving scheme and a standard finite difference scheme, showing a difference in satisfying the energy law. Numerical results are presented for validating the orders of convergence in both time and space of the new scheme for the PNP system.

Appendix: The local truncation error of the energy-preserving scheme (23)

In this “Appendix: The local truncation error of the energy-preserving scheme (23),” we show that the local truncation error of our scheme (23) is of second order in both space and time to the Nernst–Planck Eq. (5a) with the chemical potential defined in (2).

First, we define the functions $\mu _{i,-}$ and $\mu _{i,+}$ corresponding to the discrete counterparts in (21) and (22), respectively,

$$\begin{aligned}&\mu _{i,-}(x,t) {:=} k_\mathrm{B} T \left[ \log \frac{c_{i}(x,t)}{c_{i,0}} + c_{i}(x,t) g_1 (x,t) \right] \nonumber \\&\qquad + z_i e \phi (x,t), \end{aligned}$$

(53)

$$\begin{aligned}&\mu _{i,+}(x,t) {:=} k_\mathrm{B} T \left[ \log \frac{c_{i}(x,t)}{c_{i,0}} + c_{i}(x,t) g_1(x,t+\Delta t) \right] \nonumber \\&\qquad + z_i e \phi (x,t), \end{aligned}$$

(54)

where $g_1$ is defined by

$$\begin{aligned} g_1(x,t) {:=} \frac{\log (\frac{c_{i}(x,t)}{c_{i,0}})-\log (\frac{c_{i}(x,t-\Delta t)}{c_{i,0}})}{c_{i}(x,t)-c_{i}(x,t-\Delta t)}. \end{aligned}$$

(55)

Then, we build upon these functions by defining

$$\begin{aligned}&g(x,t +\frac{\Delta t}{2}):= \nonumber \\&\quad \frac{A(x)D_i(x)}{k_\mathrm{B}T}\nonumber \\&\quad \frac{c_i(x,t\!+\!\Delta t)\nabla _{\Delta x}\mu _{i,-}(x,t\!+\!\Delta t)\!+c_i(x,t)\nabla _{\Delta x}\mu _{i,+}(x,t)}{2}.\nonumber \\ \end{aligned}$$

(56)

Using Taylor expansion, one can easily show that

$$\begin{aligned}&g_1(x,t) = \frac{1}{c_{i} (x,t)} + \frac{\Delta t}{2c^2_i(x,t)} + \frac{g_2(x,t)}{k_\mathrm{B}Tc_i(x,t)} (\Delta t)^2, \nonumber \\\end{aligned}$$

(57)

$$\begin{aligned}&g_1(x,t+\Delta t) = \frac{1}{c_{i} (x,t)} - \frac{\Delta t}{2c^2_i(x,t)} + \frac{g_3(x,t)}{k_\mathrm{B}Tc_i(x,t)}(\Delta t)^2, \nonumber \\ \end{aligned}$$

(58)

where $g_2$ and $g_3$ are smooth functions depending on the derivatives of $c_i$ with respect to time t.

Substituting (57) and (58) into (53) and (54), we obtain

$$\begin{aligned} \mu _{i,-}(x,t)= & {} \mu _i(x,t)+ \frac{k_\mathrm{B} T}{2c_i(x,t)}\Delta t + g_2(x,t) (\Delta t)^2, \end{aligned}$$

(59)

$$\begin{aligned} \mu _{i,+}(x,t)= & {} \mu _i(x,t) -\frac{k_\mathrm{B} T}{2c_i(x,t)}\Delta t + g_3(x,t) (\Delta t)^2. \end{aligned}$$

(60)

By replacing $\mu _{i,+}$ and $\mu _{i,-}$ in (56) by (59) and (60), we obtain

$$\begin{aligned} g(x,t+\frac{\Delta t}{2})= & {} \frac{A(x)D_i(x)}{2k_\mathrm{B}T}\left[ (g_4(x,t+\Delta t)+g_4(x,t)) \right. \nonumber \\&+\Delta t \left( g_5(x,t+\Delta t) - g_5(x,t)\right) \nonumber \\&\qquad (\Delta t)^2 (c_i(x,t+\Delta t)\nabla _{\Delta x} g_2(x,t+\Delta t)\nonumber \\&\qquad \left. + c_i(x,t)\nabla _{\Delta x} g_3(x,t)) \right] , \end{aligned}$$

(61)

where the smooth functions $g_4$ and $g_5$ are defined as

$$\begin{aligned} g_4(x,t)= & {} c_i(x,t) \nabla _{\Delta x} \mu _i(x,t), \end{aligned}$$

(62)

$$\begin{aligned} g_5(x,t)= & {} c_i(x,t) \nabla _{\Delta x} \left( \frac{k_\mathrm{B}T}{2c_i(x,t)}\right) . \end{aligned}$$

(63)

Noting the central differencing operator $\nabla _{\Delta x}$ has the truncation error $\displaystyle {\nabla _{\Delta x} f(x) = f^{\prime }(x) + O(\Delta x)^2}$ and from the Taylor’s expansions centered at $(x,t+\frac{\Delta t}{2})$, we have

$$\begin{aligned} g(x,t+\frac{\Delta t}{2}) = \frac{A(x)D_i(x)}{k_\mathrm{B}T} \left( c_i\frac{\partial \mu _i}{\partial x}\right) (x,t+\frac{\Delta t}{2}) \nonumber \\ + g_{6}(x,t+\frac{\Delta t}{2})(\Delta x)^2 + g_{7}(x,t+\frac{\Delta t}{2})(\Delta t)^2, \end{aligned}$$

(64)

where $g_{6}$ and $g_{7}$ are smooth functions of the derivatives of the unknowns $c_i$’s and $\phi$ and the grid size $(\Delta x, \Delta t)$.

Therefore,

$$\begin{aligned}&\nabla _{\Delta x} g(x,t+\frac{\Delta t}{2}) = \frac{\partial }{\partial x}\left( \frac{AD_i}{k_\mathrm{B}T}c_i\frac{\partial \mu _i}{\partial x}\right) (x,t+\frac{\Delta t}{2}) \nonumber \\&\quad +\frac{\partial g_{6}}{\partial x}(x,t\!+\!\frac{\Delta t}{2}) (\Delta x)^2 \!+\! \frac{\partial g_{7}}{\partial x}(x,t\!+\!\frac{\Delta t}{2}) (\Delta t)^2 +\! g_8(x,t), \nonumber \\ \end{aligned}$$

(65)

where $g_8(x,t)= o((\Delta x)^2)$ and $g_8(x,t)= o((\Delta t)^2)$. Thus, the truncation error of the scheme (23) is

$$\begin{aligned}&A(x_j)\frac{c_i(x_j,t_{n+1})-c_i(x_j,t_{n})}{\Delta t} - \nabla _{\Delta x} g(x_j,t_{n+\frac{1}{2}}) \nonumber \\&\qquad = O(\Delta x)^2 +O(\Delta t)^2 \end{aligned}$$

(66)

where we have used (65) and the Nernst–Planck Eq. (5a).