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A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson–Nernst–Planck equations

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Abstract

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme.

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Correspondence to Jingwei Hu.

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This research was supported by NSF Grant DMS-1620250 and NSF CAREER Grant DMS-1654152.

A Proof of Theorem 3.5

A Proof of Theorem 3.5

In this appendix, we provide the complete proof of Theorem 3.5.

Define the set Y as

$$\begin{aligned} Y:= \left\{ \{u^{(i)}\}_{i=1}^m \Big | \ u^{(i)} \in L^2(\Omega ),\ 1\le i\le m \right\} . \end{aligned}$$
(1.115)

We construct a mapping T from Y to Y such that \(T :\overline{\mathbf{y}}:=\{{\overline{u}}^{(i)}\}_{i=1}^m \rightarrow \mathbf{y}:=\{u^{(i)}\}_{i=1}^m\), where \(\mathbf{y}\) is the weak solution of the system decoupled through given \(\overline{\mathbf{y}}\)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle -\Delta \overline{\psi } =\sum _{i=1}^m z_i {\overline{u}}^{(i)} + \rho , &{}x\in \Omega ,\\ \displaystyle \nabla \overline{\psi }\cdot \mathbf{n}= 0,&{} x \in \partial \Omega . \end{array}\right. } \end{aligned}$$
(1.116)
$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle \frac{u^{(i)} - c^{(i),n}}{\Delta t} = \Delta u^{(i)} + \frac{1}{2}\nabla \cdot \left( z_i u^{(i)} \nabla (\psi ^n+{\overline{\psi }}) \right) , &{}x\in \Omega ,\\ \displaystyle \left( \nabla u^{(i)} + \frac{1}{2} z_i u^{(i)} \nabla \left( \psi ^n+\overline{\psi }\right) \right) \cdot \mathbf{n} =0, &{}x\in \partial \Omega . \end{array}\right. } \end{aligned}$$
(1.117)

We will show that the mapping T is well defined on an appropriate space \({\mathcal {N}}\) with the following three properties:

  • self-mapping: T maps \({\mathcal {N}}\) into itself;

  • continuity: T is continuous on \({\mathcal {N}}\);

  • precompact: the image \(T({\mathcal {N}})\) is precompact (its closure is compact).

Then the existence of a fixed point is given by the Schauder’s Fixed Point Theorem, i.e. \(T (\overline{\mathbf{y}}) = \overline{\mathbf{y}} \in {\mathcal {N}}\) exists. This fixed point is a weak solution of the coupled system (3.85), (3.86).

Let \({\mathcal {N}}\) be a subset of Y that

$$\begin{aligned} {\mathcal {N}} :=\left\{ \mathbf{y} \in Y \Big | \ \sum _{i=1}^m \left\| u^{(i)}\right\| _2\le R <\infty \right\} , \end{aligned}$$

where R is some constant to be fixed later on. We claim that T maps \({\mathcal {N}}\) into itself when \(\Delta t >0\) is small enough and \(R>0\) is large enough. Multiplying the NP equation (1.117) by \(u^{(i)} \Delta t \) and integrating with respect to x yields

$$\begin{aligned}&\frac{1}{2} \left\| u^{(i)}\right\| _2^2 + \frac{1}{2} \left\| u^{(i)}-c^{(i),n}\right\| _2^2 - \frac{1}{2} \left\| c^{(i),n}\right\| _2^2 + \Delta t \left\| \nabla u^{(i)}\right\| _2^2\nonumber \\&\quad = -\, \frac{z_i \Delta t}{2} \int _{\Omega } u^{(i)} \nabla u^{(i)} \nabla (\psi ^n+{\overline{\psi }}) \ \mathrm{d} x, \end{aligned}$$
(1.118)

where the terms on the left are obtained using integration by parts and no-flux boundary condition. Applying the Hölder’s inequality, the right hand side has the estimate

$$\begin{aligned}&- \frac{z_i \Delta t}{2} \int _{\Omega } u^{(i)} \nabla u^{(i)} \nabla (\psi ^n+{\overline{\psi }}) \ \mathrm{d} x \nonumber \\&\quad \le \frac{|z_i| \Delta t}{2} \left\| u^{(i)}\right\| _3 \left\| \nabla u^{(i)}\right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}\right) \right\| _6. \end{aligned}$$
(1.119)

To estimate the right hand side, we introduce the following \(L_p\) interpolation inequality (a special case of Gagliardo–Nirenberg interpolation inequality)

$$\begin{aligned} \left\| v\right\| _{3} \le K_1 \left\| v\right\| _{2}^{1-\mu } \left\| \nabla v\right\| _{2}^{\mu },\quad \mu ={\left\{ \begin{array}{ll} 1/2, &{} \text {when }d=3,\\ 1/3, &{} \text {when }d=2,\\ 1/6, &{} \text {when }d=1, \end{array}\right. } \end{aligned}$$
(1.120)

and the following Sobolev embedding theorem \((d\le 3)\) for \(v\in W^{1,2}_0(\Omega )\)

$$\begin{aligned} \left\| v \right\| _6 \le K_2 \left\| \nabla v\right\| _{2}, \end{aligned}$$
(1.121)

where \(K_1\) and \(K_2\) are constants.

From (1.120) and (1.121), the right hand side of (1.119) is now estimated as

$$\begin{aligned}&\frac{|z_i| \Delta t}{2} \left\| u^{(i)}\right\| _3 \left\| \nabla u^{(i)}\right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}\right) \right\| _6 \nonumber \\&\quad \le \frac{|z_i| \Delta t K}{2} \left\| u^{(i)}\right\| _2^{1-\mu } \left\| \nabla u^{(i)}\right\| _2 ^{1+\mu } \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2 \nonumber \\&\quad \le \frac{|z_i| \Delta t K}{2} \left( \frac{\left\| \nabla u^{(i)}\right\| _2 ^{2}}{\frac{2}{1+\mu }\gamma ^{\frac{2}{1+\mu }}} + \frac{\gamma ^{\frac{2}{1-\mu }} \left\| u^{(i)}\right\| _2^{2} \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1-\mu }}}{2/(1-\mu )} \right) ,\nonumber \\ \end{aligned}$$
(1.122)

for some constant K, where the last line follows Young’s inequality.

Choosing \(\gamma = \left( \frac{(1+\mu )|z_i|K}{4}\right) ^{\frac{1+\mu }{2}}\) gives

$$\begin{aligned}&\frac{|z_i| \Delta t}{2} \left\| u^{(i)}\right\| _3 \left\| \nabla u^{(i)}\right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}\right) \right\| _6 \nonumber \\&\quad \le \Delta t \left\| \nabla u^{(i)}\right\| _2 ^{2} + \Delta t \frac{(1-\mu )|z_i|K}{4} \left( \frac{(1+\mu )|z_i|K}{4}\right) ^{\frac{1+\mu }{1-\mu }} \left\| u^{(i)}\right\| _2^{2}\nonumber \\&\qquad \times \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1-\mu }}. \end{aligned}$$
(1.123)

Combining (1.118), (1.119) and (1.123), we get

$$\begin{aligned}&\frac{1}{2} \left\| u^{(i)}\right\| _2^2 + \frac{1}{2} \left\| u^{(i)}-c^{(i),n}\right\| _2^2 - \frac{1}{2} \left\| c^{(i),n}\right\| _2^2 \nonumber \\&\quad \le \Delta t \frac{(1-\mu )|z_i|K}{4} \left( \frac{(1+\mu )|z_i|K}{4}\right) ^{\frac{1+\mu }{1-\mu }}\nonumber \\&\quad \left\| u^{(i)}\right\| _2^{2} \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1-\mu }}, \end{aligned}$$
(1.124)

which is

$$\begin{aligned}&\left( 1 - \Delta t \frac{(1-\mu )|z_i|K}{2} \left( \frac{(1+\mu )|z_i|K}{4}\right) ^{\frac{1+\mu }{1-\mu }}\right. \nonumber \\&\quad \left. \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1-\mu }} \right) \left\| u^{(i)}\right\| _2^{2} \le \left\| c^{(i),n}\right\| _2^2. \end{aligned}$$
(1.125)

Thus, if \(\Delta t>0\) is sufficient small and \(R>0\) is sufficient large, we can conclude that

$$\begin{aligned} \sum _{i=1}^m \left\| u^{(i)}\right\| _2^{2} \le R. \end{aligned}$$
(1.126)

In other words, T maps \({\mathcal {N}}\) into itself.

Similar to (1.125), we can get the estimate for \(H^1\) norm of \(u^{(i)}\) that

$$\begin{aligned}&\frac{|z_i| \Delta t}{2} \left\| u^{(i)}\right\| _3 \left\| \nabla u^{(i)}\right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}\right) \right\| _6 \nonumber \\&\quad \le \frac{|z_i| \Delta t K}{2} \left\| u^{(i)}\right\| _2^{1-\mu } \left\| \nabla u^{(i)}\right\| _2 ^{1+\mu } \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2 \nonumber \\&\quad \le \frac{|z_i| \Delta t K}{2} \left( \frac{\left\| u^{(i)}\right\| _2 ^{2}}{\frac{2}{1-\mu }\gamma ^{\frac{2}{1-\mu }}} + \frac{\gamma ^{\frac{2}{1+\mu }} \left\| \nabla u^{(i)}\right\| _2^{2} \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1+\mu }}}{2/(1+\mu )} \right) .\nonumber \\ \end{aligned}$$
(1.127)

Choosing \(\gamma = \left( \frac{(1-\mu )|z_i|\Delta t K}{2}\right) ^{\frac{1-\mu }{2}}\) gives

$$\begin{aligned}&\frac{|z_i| \Delta t}{2} \left\| u^{(i)}\right\| _3 \left\| \nabla u^{(i)}\right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}\right) \right\| _6 \nonumber \\&\quad \le \frac{1}{2} \left\| u^{(i)}\right\| _2 ^{2} + \frac{(1+\mu )|z_i|\Delta t K}{4} \left( \frac{(1-\mu )|z_i|\Delta t K}{2}\right) ^{\frac{1-\mu }{1+\mu }} \left\| \nabla u^{(i)}\right\| _2^{2}\nonumber \\&\quad \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1+\mu }}. \end{aligned}$$
(1.128)

Combining (1.118), (1.119) and (1.128), we get

$$\begin{aligned}&\frac{1}{2} \left\| u^{(i)}-c^{(i),n}\right\| _2^2 - \frac{1}{2} \left\| c^{(i),n}\right\| _2^2 + \Delta t \left\| \nabla u^{(i)}\right\| _2^2\nonumber \\&\quad \le \frac{(1+\mu )|z_i|\Delta t K}{4} \left( \frac{(1-\mu )|z_i|\Delta t K}{2}\right) ^{\frac{1-\mu }{1+\mu }} \left\| \nabla u^{(i)}\right\| _2^{2}\nonumber \\&\quad \left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1+\mu }}, \end{aligned}$$
(1.129)

which is

$$\begin{aligned}&\left( 1-\frac{(1+\mu )|z_i| K}{4} \left( \frac{(1-\mu )|z_i|\Delta t K}{2}\right) ^{\frac{1-\mu }{1+\mu }}\left\| \sum _{i=1}^m z_i \left( c^{(i),n} +{\overline{u}}^{(i)}\right) + 2 \rho \right\| _2^{\frac{2}{1+\mu }}\right) \nonumber \\&\quad \left\| \nabla u^{(i)}\right\| _2^2\le \frac{1}{2 \Delta t} \left\| c^{(i),n}\right\| _2^2. \end{aligned}$$
(1.130)

Note that if \(\Delta t\) is sufficient small, we get

$$\begin{aligned} \sum _{i=1}^m \left\| \nabla u^{(i)}\right\| _2^{2} \le R', \end{aligned}$$
(1.131)

here \(R'\) is some constant. Then we can say that each component of \(\mathbf{y}\) belongs to \( H^1(\Omega )\).

Next, to prove the continuity of T, we define \(T: \overline{\mathbf{y}}_l:=\{{\overline{u}}_l^{(i)}\}_{i=1}^m \rightarrow \mathbf{y}_l:=\{u_l^{(i)}\}_{i=1}^m\) for \(l=1,2\). Then the NP equation (1.117) for \(\mathbf{y}_l\) gives

$$\begin{aligned} \frac{1}{\Delta t}\left( u_1^{(i)} -u_2^{(i)}\right)= & {} \Delta \left( u_1^{(i)} -u_2^{(i)}\right) + \frac{1}{2}\nabla \left( z_i \left( u_1^{(i)} -u_2^{(i)}\right) \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right) \nonumber \\&+\frac{1}{2}\nabla \left( z_i u_2^{(i)} \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \right) . \end{aligned}$$
(1.132)

Multiplying by \( \left( u_1^{(i)} -u_2^{(i)}\right) \) and integrating with respect to x leads to

$$\begin{aligned}&\frac{1}{\Delta t}\left\| u_1^{(i)} -u_2^{(i)}\right\| _2^2 + \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2^2 \nonumber \\&\quad = -\,\frac{z_i}{2}\int \left( u_1^{(i)} - u_2^{(i)}\right) \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \cdot \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \mathrm{d}x \nonumber \\&\qquad - \frac{z_i}{2} \int u_2^{(i)} \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \cdot \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \mathrm{d}x \nonumber \\&\quad \le \frac{|z_i|}{2} \left( \left\| u_1^{(i)} - u_2^{(i)}\right\| _3 \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6 \right. \nonumber \\&\qquad \left. +\left\| u_2^{(i)} \right\| _3 \left\| \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \right\| _6 \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2 \right) , \end{aligned}$$
(1.133)

the last line follows Hölder’s inequality and boundary integral vanishes because of the non-flux condition. By \(L_p\) interpolation (1.120) and Young’s inequality with parameter \(\gamma =\left( \frac{(1+\mu )|z_i|K}{2}\right) ^{\frac{1+\mu }{2}}\), the estimate for first term is given as

$$\begin{aligned}&\frac{|z_i|}{2} \left\| u_1^{(i)} - u_2^{(i)}\right\| _3 \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6 \nonumber \\&\quad \le \frac{|z_i|K}{2} \left\| u_1^{(i)} - u_2^{(i)}\right\| _2^{1-\mu } \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2 ^{1+\mu } \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6\nonumber \\&\quad \le \frac{|z_i|K}{2} \left( \frac{\left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2^2}{\frac{2}{1+\mu }\gamma ^{\frac{2}{1+\mu }}} +\frac{\gamma ^{\frac{2}{1-\mu }}\left\| u_1^{(i)} - u_2^{(i)}\right\| _2^2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6^{\frac{2}{1-\mu }}}{2/(1-\mu )}\right) \nonumber \\&\quad =\frac{1}{2}{\left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2^2}\nonumber \\&\quad +\frac{(1-\mu )|z_i|K}{4} \left( \frac{(1+\mu )|z_i|K}{2}\right) ^{\frac{1+\mu }{1-\mu }} \left\| u_1^{(i)} - u_2^{(i)}\right\| _2^2 \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6^{\frac{2}{1-\mu }}.\nonumber \\ \end{aligned}$$
(1.134)

For the second term, by Cauchy inequality, we have

$$\begin{aligned}&\frac{|z_i|}{2}\left\| u_2^{(i)} \right\| _3 \left\| \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \right\| _6 \left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2 \nonumber \\&\quad \le \frac{1}{2} {\left\| \nabla \left( u_1^{(i)} - u_2^{(i)}\right) \right\| _2^2} +\frac{|z_i|^2 \left\| u_2^{(i)} \right\| _3^2 \left\| \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \right\| _6^2}{8}. \end{aligned}$$
(1.135)

Hence we arrive at the following estimate

$$\begin{aligned}&\left( \frac{1}{\Delta t} -\frac{(1-\mu )|z_i|K}{4} \left( \frac{(1+\mu )|z_i|K}{2}\right) ^{\frac{1+\mu }{1-\mu }} \left\| \nabla \left( \psi ^n+{\overline{\psi }}_1\right) \right\| _6^{\frac{2}{1-\mu }} \right) \left\| u_1^{(i)} -u_2^{(i)}\right\| _2^2\nonumber \\&\quad \le \frac{1}{8}|z_i|^2\left\| u_2^{(i)} \right\| _3^2 \left\| \nabla \left( {\overline{\psi }}_1-{\overline{\psi }}_2\right) \right\| _6^2 \nonumber \\&\quad \le \frac{K}{8} |z_i|^2\left\| u_2^{(i)} \right\| _2^{2-2\mu } \left\| \nabla u_2^{(i)} \right\| _2^{2\mu } \left\| \sum _{i=1}^m z_i \left( \overline{u}_1^{(i)}-\overline{u}_2^{(i)}\right) \right\| _2^2, \end{aligned}$$
(1.136)

the last inequality follows the \(L_p\) interpolation (1.120) and Sobolev embedding (1.121). Therefore, the difference \(\left\| u_1^{(i)} -u_2^{(i)}\right\| _2\) can be controlled via \(\left\| \overline{u}_1^{(i)} - \overline{u}_2^{(i)}\right\| _2\) when \(\Delta t\) is sufficient small. In other words, we proved the uniformly Lipschitz continuity of mapping T.

Note that \({\mathcal {N}}\) is a closed and convex subset of Y. The precompact property of \(T({\mathcal {N}})\) is an immediate consequence of the continuity and estimate (1.126).

Thus, the existence of a fixed point in \({\mathcal {N}}\) is guaranteed by the Schauder’s Fixed Point Theorem. Therefore, we proved the existence of weak solution to the system (3.85), (3.86) for \(\Delta t >0\) small enough, which belongs to \(H^1(\Omega )\).

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Hu, J., Huang, X. A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson–Nernst–Planck equations. Numer. Math. 145, 77–115 (2020). https://doi.org/10.1007/s00211-020-01109-z

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