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The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to…

Computational Physics · Physics 2025-12-24 Michal Bosy , Matthew W. Scroggs , Timo Betcke , Erik Burman , Christopher D. Cooper

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit…

Numerical Analysis · Mathematics 2018-04-04 Yimin Zhong , Kui Ren , Richard Tsai

An accurate force calculation with the Poisson-Boltzmann equation is challenging, as it requires the electric field on the molecular surface. Here, we present a calculation of the electric field on the solute-solvent interface that is exact…

Chemical Physics · Physics 2023-01-13 Ian Addison-Smith , Horacio V. Guzmán , Christopher D. Cooper

The Poisson-Boltzmann equation is a widely used model to study the electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate…

Numerical Analysis · Mathematics 2020-09-25 Vicente Ramm , Jehanzeb H. Chaudhry , Christopher D. Cooper

In this paper, we present a parallel higher-order boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov…

Numerical Analysis · Mathematics 2015-06-12 Weihua Geng

The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into…

Materials Science · Physics 2016-01-20 G. Fisicaro , L. Genovese , O. Andreussi , N. Marzari , S. Goedecker

We describe a three-stage procedure to analyze the dependence of Poisson Boltzmann calculations on the shape, size and geometry of the boundary between solute and solvent. Our study is carried out within the boundary element formalism, but…

Biological Physics · Physics 2007-11-27 P. Kar , Y. Wei , U. H. E. Hansmann , S. Hoefinger

The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method…

Computational Physics · Physics 2021-10-27 Jiahui Chen , Johannes Tausch , Weihua Geng

Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges, however, other force fields…

Computational Physics · Physics 2018-10-10 Christopher D. Cooper

In a continuum model of the solvation of charged molecules in an aqueous solvent, the classical Poisson-Boltzmann (PB) theory is generalized to include the solute point charges and the dielectric boundary that separates the high-dielectric…

Optimization and Control · Mathematics 2021-09-29 Bo Li , Zhengfang Zhang , Shenggao Zhou

The Poisson-Boltzmann equation is widely used to model molecular electrostatics; however, it is usually solved in linearised form because the sinh nonlinearity is challenging, limiting its applicability in highly charged systems such as…

Computational Physics · Physics 2026-04-20 Mauricio Guerrero-Montero , Michal Bosy , Christopher D. Cooper

We present the theory and implementation of a Poisson-Boltzmann implicit solvation model for electrolyte solutions. This model can be combined with arbitrary electronic structure methods that provide an accurate charge density of the…

Computational Physics · Physics 2020-01-08 Christopher J. Stein , John M. Herbert , Martin Head-Gordon

A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is…

Numerical Analysis · Mathematics 2010-01-12 Long Chen , Michael Holst , Jinchao Xu

Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential…

Numerical Analysis · Mathematics 2024-09-26 Xiaofeng Xu , Lian Zhang , Yin Shi , Long-Qing Chen , Jinchao Xu

This review describes the theory and implementation of implicit solvation models based on continuum electrostatics. Within quantum chemistry this formalism is sometimes synonymous with the polarizable continuum model, a particular…

Chemical Physics · Physics 2022-03-15 John M. Herbert

The Poisson-Boltzmann (PB) model governs the electrostatics of solvated biomolecules, i.e., potential, field, energy, and force. These quantities can provide useful information about protein properties, functions, and dynamics. By…

Biomolecules · Quantitative Biology 2023-12-29 Xin Yang , Elyssa Sliheet , Reece Iriye , Daniel Reynolds , Weihua Geng

Electrostatic forces play many important roles in molecular biology, but are hard to model due to the complicated interactions between biomolecules and the surrounding solvent, a fluid composed of water and dissolved ions. Continuum model…

Numerical Analysis · Mathematics 2015-12-29 Jaydeep P. Bardhan , Matthew G. Knepley

We consider the simulation of electromagnetic scattering by single and multiple isotropic homogeneous dielectric particles using boundary integral equations. Galerkin discretizations of the classical Poggio-Miller-Chang-Harrington-Wu-Tsai…

Numerical Analysis · Mathematics 2020-05-14 Antigoni Kleanthous , Timo Betcke , David P. Hewett , Matthew W. Scroggs , Anthony J. Baran

This paper proposes a collocation boundary element method based on the Burton--Miller method for solving transmission problems, which is rapidly convergent within the Krylov subspace solver framework. Our study enhances Burton--Miller-type…

Numerical Analysis · Mathematics 2025-01-17 Yasuhiro Matsumoto , Akihiro Yoshiki , Hiroshi Isakari

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems,…

Numerical Analysis · Mathematics 2025-12-16 Kai Weixian Lan , Elias Gueidon , Ayano Kaneda , Julian Panetta , Joseph Teran
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