在數值方法解泊松-斯特-普朗克模型發展中，凡得瓦爾（vdW）位能的介電常數裡玻茲曼分佈是一個連續函數。凡得瓦爾的位能表示為在溶劑中的離子和原子在生物大分子（蛋白質）所有成對的蘭納-瓊斯之間的相互作用的總和，並在周圍的分子或原子內部產生位能牆。在本篇中我們發現直接使用差分法處理位能牆的近似解較為不理想。

Numerical methods are developed for solving the Poisson-Nernst-Planck model in which the electric permittivity is a continuous function of the Boltzmann distribution in terms of the van der Waals (vdW) potential. The vdW potential is expressed as a summation of all pairwise the Lennard-Jones interactions between ions in solvent and the atoms in a biomolecule (protein). The vdW potential has internal layer (potential wall) around the molecule. It is found that direct finite difference approximation of the vdW potential is unable to capture the potential wall but gives good approximation away from the wall. On the other hand, a splitting function method can yield a sharp wall but does not give good approximation away from the wall.

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