泊松波茲曼方程是一個非線性偏微分方程，它描述了分子和溶劑之間的靜電相的互作用。此方程在分子動力學和生物物理領域往往很重要，且在複雜系統中是難以解決的。Kirkwood的介質球提供了一個優秀的基準去測試泊松 - 玻爾茲曼方程求解的精度、收斂速度和效率。然而，在蛋白質分子表面要達到二階精度，我們必須處理奇異的電荷和界面問題。我們使用由陳、劉、王提出的分解方法[2]去應對奇異電荷，以及使用匹配接口和邊界的MIB方法處理接口的問題。

The Poisson—Boltzmann equation is a non-linear partial differential equation. It describes the electrostatic interactions between molecule and solvent. The equation
is important in the fields of molecular dynamics and biophysics. And it is often difficult to solve for complex systems. Kirkwood’s dielectric sphere provides an excellent
benchmark for testing Poisson-Boltzmann (PB) solvers in terms of accuracy, speed of convergence, and efficiency [8]. However, to achieve second order accuracy for the molecular surfaces of proteins, we have to treat singular charge and interface problem. The decomposition method proposed by Chern, Liu, and Wang [2] cope with the singular charges, and the MIB (matched interface and boundary) method [19] for interface problem.

[1] N. A. Baker, Improving implicit solvent simulations: a Poisson-centric view, Curr.
Opin. Struct. Biol. 15, 137 2005 .
[2] I-L. Chern, J.-G. Liu, and W.-C. Wang, Accurate evaluation of electrostatics for
macromolecules in solution, Methods Appl. Anal. 10 (2003) 309–328.
[3] C. M. Cortis and R. A. Friesner, Numerical solution of the Poisson—Boltzmann equation
using tetrahedral finite-element meshes, J. Comput. Chem. 18, 1591 (1997) .
[4] F. Fogolari, A. Brigo, and H. Molinari, The Poisson—Boltzmann equation for biomolecular
electrostatics: a tool for structural biology, J. Mol. Recognit. 15, 377 (2002).
[5] M. Feig and C. L. Brooks III, Recent advances in the development and application
of implicit solvent models in biomolecule simulations, Curr. Opin. Struct. Biol. 14,
217 (2004)
[6] K. E. Forsten, R. E. Kozack, D. A. Lauffenburger, and S. Subramaniam, J. Phys.
Chem. 98, 5580 (1994)
[7] F. Fogolari, A. Brigo, and H. Molinari, Protocol for MM/PBSA Molecular Dynamics
Simulations of Proteins, Biophys. J. 85, 159 (2003) .
[8] W. Geng, S. Yu, and G. Weia, Treatment of charge singularities in implicit solvent
models, J. Chem. Phys. 127 (2007) 114106.
[9] B. Honig and A. Nicholls, Classical electrostatics in biology and chemistry , Science
268, 1144 (1995) .
[10] P. A. Kollman, I. Massova, and C. Reyes et al. Calculating Structures and Free Energies
of Complex Molecules:Combining Molecular Mechanics and Continuum Models,
Acc. Chem. Res. 33, 889 (2000) .
[11] J.-L. Liu, Lecture Notes on Poisson-Nernst-Planck Modeling and Simulation of Biological
Ion Channels, 2012.
[12] C. S. Peskin. Numerical analysis of blood flow in the heart. Journal of Computational
Physics, 25(3):220—52, 1977.
[13] B. Roux and T. Simonson, Implicit solvent models, Biophys. Chem. 78, 1 (1999) .
[14] K. A. Sharp and B. Honig, Electrostatic Interactions in Macromolecules: Theory and
Applications, Annu. Rev. Biophys. Biophys. Chem. 19, 301 (1990) .
[15] J. M. J. Swanson, R. H. Henchman, and J. A. McCammon, Revisiting Free Energy
Calculations: A Theoretical Connection to MM/PBSA and Direct Calculation of the
Association Free Energy, Biophys. J.86, 67 (2004).
[16] Y. N. Vorobjev and H. A. Scheraga, A fast adaptive multigrid boundary element
method for macromolecular electrostatic computations in a solvent , J Comput.
Chem. 18, 569 (1997) .
[17] J. Warwicker and H. C. Watson, Calculation of the electric potential in the active
site cleft due to 0-helix dipoles, J. Mol. Biol. 154, 671 (1982)
[18] Y. Yanga, A. Zhou, A finite element recovery approach to Green’s function approximations
with applications to electrostatic potential computation, J. Comput. Appl.,
225 (2009), pp. 202—212.
[19] Q. Zheng, D. Chen, and G.-W. Wei, Second-order Poisson Nernst-Planck solver for
ion channel transport, J. Comp. Phys. 230 (2011) 5239-5262.
[20] R. J. Zauhar and R. S. Morgan, A new method for computing the macromolecular
electric potential, J. Mol. Biol. 186, 815 (1985) .