Positivity-Preserving Hybridizable Discontinuous Galerkin Scheme for Solving PNP Model

Figures

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  Figure 1. The parameter values used for Example 2. (a) Plot of r(x). (b) Plot of A=πr(x)2. (c) Plot of ε(x). (d) Plot of ρ0(x).

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  Figure 2. Solution ϕh for Equation (22) (Example 2).

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  Figure 3. Estimated solution u1,h and c1,h for Equation (22), where c1,h=exp(u1,h). (a) Numerical solution u1,h for Equation (22). (b) Numerical solution c1,h for Equation (22). This density solution develops a large spike, and the solution maintains positive (Example 2).

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  Figure 4. Estimated solution u2,h and c2,h for Equation (22), where c2,h=exp(u2,h). (a) Numerical solution u2,h for Equation (22). (b) Numerical solution c2,h for Equation (22) (Example 2).

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  Figure 5. Solution ϕh for Equation (22) (Example 3). Notice that there is almost no noticeable difference between this result and result in Figure 2. The second-order time discretization does not significantly improve the solution resolution.

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  Figure 6. Estimated solution u1,h and c1,h for Equation (22), where c1,h=exp(u1,h). (a) Numerical solution u1,h for Equation (22). (b) Numerical solution c1,h for Equation (22). No negative density is observed (Example 3). Notice that there is slight difference between this result and result in Figure 3. The second-order time discretization does not significantly improve the solution resolution.

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  Figure 7. Estimated solution u2,h and c2,h for Equation (22), where c2,h=exp(u2,h). (a) Numerical solution u2,h for Equation (22). (b) Numerical solution c2,h for Equation (22). No negative density is observed (Example 3). Notice that there is slight difference between this result and result in Figure 2. The second-order time discretization does not significantly improve the solution resolution.

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  Figure 8. Ion Channel Geometry: Coordinates of the domain Ω are: A:(−28,7), B:(−18,2), C:(−5,2), D:(−5,0.5), E:(10,0.5), F:(25,14), L:(−28,0), R:(25,0) (Example 4).

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  Figure 9. Solution ϕh for Equation (23) (Example 4).

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  Figure 10. (a) Numerical solution u1,h for Equation (23) at t=1. (b) Numerical solution c1,h for Equation (23) at t=1. Positivity of density is preserved (Example 4).

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  Figure 11. (a) Numerical solution u2,h for Equation (23) at t=1. (b) Numerical solution c2,h for Equation (23) at t=1. Positivity of density is preserved (Example 4).

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  Figure 12. Numerical solution ϕh for Equation (23) cut at y=0 (Example 4).