Abstract
A recent article by Deng and Cai introduced fourth-order image approximations to the reaction field for a charge inside a dielectric sphere immersed in a solvent of low ionic strength (J. Comput. Phys. 227:1246–1266, 2007). To represent such a reaction field, the image approximations employ a point charge at the classical Kelvin image point and two line charges that extend from the Kelvin image point along the radial direction to infinity. In this paper, a sixth-order image approximation is developed, using the same point charge with three different line charges. Procedures on how to discretize the line charges by point image charges and how to implement the resulting point image approximation in O(N) complexity for potential and force field calculations are included. Numerical results demonstrate the sixth-order convergence rate of the image approximation and the O(N) complexity of the fast implementation of the point image approximation.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Cai, W., Deng, S., Jacobs, D.: Extending the fast multipole method to charges inside or outside a dielectric sphere. J. Comput. Phys. 223, 846–864 (2007)
Carrier, J., Greengard, L.F., Rokhlin, V.: A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput. 9, 669–686 (1988)
Cheng, H., Greengard, L.F., Roklin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155, 468–498 (1999)
Deng, S., Cai, W.: Discrete image approximations of ionic solvent induced reaction field to charges. Commun. Comput. Phys. 2, 1007–1026 (2007)
Deng, S., Cai, W.: Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: High-order image approximations for reaction fields. J. Comput. Phys. 227, 1246–1266 (2007)
Esselink, L.: A comparison of algorithms for long-range interactions. Comput. Phys. Commun. 87, 375–395 (1995)
Gautschi, W.: Algorithm 726: ORTHPOL—a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–62 (1994)
Gradshteyn, I.S., Ayzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, Boston (1994)
Greengard, L.F.: The Rapid Evaluation of Potential Fields in Particle Systems. MIT, Cambridge (1987)
Greengard, L.F., Huang, J.: A new version of the fast multipole method for screened Coulomb interactions in three dimensions. J. Comput. Phys. 180, 642–658 (2002)
Greengard, L.F., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)
Greengard, L.F., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6, 229–269 (1997)
Juffer, A., Botta, E.F.F., van Keulen, B.A.M., van der Ploeg, A., Berendsen, H.J.C.: The electric potential of a macromolecule in a solvent: A fundamental approach. J. Comput. Phys. 97, 144–171 (1991)
Lee, M.S., Olson, M.A.: Evaluation of Poisson solvation models using a hybrid explicit/implicit solvent method. J. Phys. Chem. B 109, 5223–5236 (2005)
Lee, M.S., Salsbury, Jr. F.R., Olson, M.A.: An efficient hybrid explicit/implicit solvent method for biomolecular simulations. J. Comput. Chem. 25, 1967–1978 (2004)
Lindell, I.V.: Electrostatic image theory for the dielectric sphere. Radio Sci. 27, 1–8 (1992)
Lu, B., Cheng, X., McCammon, J.A.: New-version-fast-multipole-method accelerated electrostatic calculations in biomolecular systems. J. Comput. Phys. 226, 1348–1366 (2007)
Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)
Neumann, C.: Hydrodynamische Untersuchen nebst einem Anhang uber die Probleme der Elecktrostatik und der magnetischen Induktion. Teubner, Leipzig, pp. 279–282 (1883)
Norris, W.T.: Charge images in a dielectric sphere. IEE Proc. Sci. Meas. Technol. 142, 142–150 (1995)
Okur, A., Simmerling, C.: Hybrid explicit/implicit solvation methods. In: Spellmeyer, D. (ed.) Annu. Rep. Comput. Chem., vol. 2, chap. 6 (2006)
Phillips, J.R., White, J.K.: A precorrected-FFT method for electrostatic analysis of complicated 3-D structures. IEEE Trans. Comput. Aided Des. Integr. Circ. Syst. 10, 1059–1072 (1997)
Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196, 591–626 (2004)
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Xiang, M., Deng, S. & Cai, W. A Sixth-order Image Approximation to the Ionic Solvent Induced Reaction Field. J Sci Comput 41, 411–435 (2009). https://doi.org/10.1007/s10915-009-9307-z
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DOI: https://doi.org/10.1007/s10915-009-9307-z