Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery
Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery
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DOI：
10.1016/j.jcp.2020.109958
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发表时间：
2020-10
期刊：
影响因子：
--
通讯作者：
Arum Lee;Weihua Geng;Shan Zhao
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文献类型：
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作者：
Arum Lee;Weihua Geng;Shan Zhao
Poisson-Boltzmann方程数值解的一个重大挑战是由于Dirac Delta函数形式的奇异电荷源。为了克服这一困难，人们发展了几种正则化方法，将势函数分解成两个或三个部分，以便可以使用格林函数来解析求解奇异分量，而其他分量则是有界的。然而，在文献中观察到，尽管一些正则化方法在分析上是等价的，但由于不清楚的原因，它们的准确性明显低于其他方法。为了理解这种差异，本文研究了四种流行的正则化方法的数值性能，并用匹配界面和边界(MIB)方法实现了它们。MIB方法是一种处理具有间断系数的椭圆界面问题的复杂有限差分方法。在四种方法都表现出二阶收敛的情况下，两种格式的精度都有下降。本文提供了数值分析和实验来追踪这种还原的来源，并将误差与格林函数的拉普拉斯函数落在蛋白质结构域之外的事实联系在一起。虽然这一项在解析上消失了，但它的数值疏忽引入了离散化误差。通过建立适当的椭圆界面问题，提出了一种有效的精度恢复技术，从而使四种方法都能产生相同的高精度。通过这项研究，所有涉及的正则化方案都得到了更好的理解，并很好地联系到了一个统一的框架中。
A significant challenge in numerical solution to the Poisson-Boltzmann equation is due to singular charge sources in terms of Dirac delta functions. To overcome this difficulty, several regularization methods have been developed, in which the potential function is decomposed into two or three parts so that the singular component can be analytically solved using the Green's function, while other components become bounded. However, it was observed in the literature that some regularization methods are significantly less accurate than the others for unclear reasons, even though they are analytically equivalent. To understand this discrepancy, the numerical performance of four popular regularization methods is investigated in this work by implementing them with the Matched Interface and Boundary (MIB) approach, which is a sophisticated finite difference method for treating elliptic interface problems with discontinuous coefficients. With all four methods showing second order convergence, accuracy reduction is numerically observed in two schemes. This paper provides numerical analysis and experiment to trace the source of such reduction, and links the error to the fact that the Laplacian of Green's function is dropped outside the protein domain. While this term is analytically vanishing, its numerical negligence introduces a discretization error. Formulating via a proper elliptic interface problem, an effective accuracy recovery technique is proposed so that all four methods yield the same high precision. With this study, all involved regularization schemes are better understood and well connected into a unified framework.