A Poisson-Boltzmann solver on irregular domains with Neumann or Robin boundary conditions on non-graded adaptive grid

We introduce a second-order solver for the Poisson-Boltzmann equation in arbitrary geometry in two and three spatial dimensions. The method differs from existing methods solving the Poisson-Boltzmann equation in the two following ways: first, non-graded Quadtree (in two spatial dimensions) and Octree (in three spatial dimensions) grid structures are used; Second, Neumann or Robin boundary conditions are enforced at the irregular domain's boundary. The irregular domain is described implicitly and the grid needs not to conform to the domain's boundary, which makes grid generation straightforward and robust. The linear system is symmetric, positive definite in the case where the grid is uniform, nonsymmetric otherwise. In this case, the resulting matrix is an M-matrix, thus the linear system is invertible. Convergence examples are given in both two and three spatial dimensions and demonstrate that the solution is second-order accurate and that Quadtree/Octree grid structures save a significant amount of computational power at no sacrifice in accuracy.

[1]  Peijuan Zhu,et al.  Implementation and testing of stable, fast implicit solvation in molecular dynamics using the smooth‐permittivity finite difference Poisson–Boltzmann method , 2004, J. Comput. Chem..

[2]  Richard A. Friesner,et al.  Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes , 1997, J. Comput. Chem..

[3]  Michael J. Aftosmis,et al.  Adaptive Cartesian mesh generation , 1998 .

[4]  Frédéric Gibou,et al.  A second order accurate level set method on non-graded adaptive cartesian grids , 2007, J. Comput. Phys..

[5]  Dongqing Li,et al.  Modeling forced liquid convection in rectangular microchannels with electrokinetic effects , 1998 .

[6]  Hanan Samet,et al.  Applications of spatial data structures - computer graphics, image processing, and GIS , 1990 .

[7]  R. Fedkiw,et al.  A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem , 2005 .

[8]  Ronald Fedkiw,et al.  A Level Set Approach for the Numerical Simulation of Dendritic Growth , 2003, J. Sci. Comput..

[9]  Ray Luo,et al.  A Poisson–Boltzmann dynamics method with nonperiodic boundary condition , 2003 .

[10]  D. Hansford,et al.  Effect of multivalent ions on electroosmotic flow in micro‐ and nanochannels , 2002, Electrophoresis.

[11]  W. Im,et al.  Continuum solvation model: Computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation , 1998 .

[12]  L. R. Scott,et al.  Electrostatics and diffusion of molecules in solution: simulations with the University of Houston Brownian dynamics program , 1995 .

[13]  Wei-Cheng Wang.  A Jump Condition Capturing Finite Difference Scheme for Elliptic Interface Problems , 2004, SIAM J. Sci. Comput..

[14]  Chang-Ock Lee,et al.  A discontinuous Galerkin method for elliptic interface problems with application to electroporation , 2009 .

[15]  Guo-Wei Wei,et al.  Highly accurate biomolecular electrostatics in continuum dielectric environments , 2008, J. Comput. Chem..

[16]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[17]  Zhonghua Qiao,et al.  A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES 1) , 2006 .

[18]  Zhong-hua Qiao.  A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES , 2006 .

[19]  Dexuan Xie,et al.  A new minimization protocol for solving nonlinear Poisson–Boltzmann mortar finite element equation , 2007 .

[20]  Frédéric Gibou,et al.  Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions , 2010, J. Comput. Phys..

[21]  Zhonghua Qiao,et al.  A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations , 2007, J. Sci. Comput..

[22]  Chohong Min.  Local level set method in high dimension and codimension , 2004 .

[23]  Frédéric Gibou,et al.  A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate , 2009, J. Comput. Phys..

[24]  P. E. Dyshlovenko.  Adaptive numerical method for Poisson-Boltzmann equation and its application , 2002 .

[25]  Barry Honig,et al.  Focusing of electric fields in the active site of Cu‐Zn superoxide dismutase: Effects of ionic strength and amino‐acid modification , 1986, Proteins.

[26]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[27]  A. Brigo,et al.  The Poisson–Boltzmann equation for biomolecular electrostatics: a tool for structural biology , 2002, Journal of molecular recognition : JMR.

[28]  J. A. McCammon,et al.  R EVIEW A RTICLE Recent Progress inNumericalMethods for the Poisson-Boltzmann Equation in Biophysical Applications , 2008 .

[29]  Frédéric Gibou,et al.  A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids , 2007, J. Sci. Comput..

[30]  Donald Bashford,et al.  An Object-Oriented Programming Suite for Electrostatic Effects in Biological Molecules , 1997, ISCOPE.

[31]  Michael Holst,et al.  Multilevel Methods for the Poisson-Boltzmann Equation , 1993 .

[32]  Frédéric Gibou,et al.  A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids , 2006, J. Comput. Phys..

[33]  Li-Tien Cheng,et al.  A second-order-accurate symmetric discretization of the Poisson equation on irregular domains , 2002 .

[34]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[35]  Frédéric Gibou,et al.  Geometric integration over irregular domains with application to level-set methods , 2007, J. Comput. Phys..

[36]  Frédéric Gibou,et al.  A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids , 2006, J. Comput. Phys..

[37]  Michael J. Holst,et al.  Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples , 2001 .

[38]  Michael J. Holst,et al.  The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers , 2001, IBM J. Res. Dev..

[39]  Michael J. Holst,et al.  The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation , 2007, SIAM J. Numer. Anal..

[40]  J. Milovich,et al.  Solution of the nonlinear Poisson-Boltzmann equation using pseudo-transient continuation and the finite element method. , 2002, Journal of colloid and interface science.

[41]  Michael J. Holst,et al.  Adaptive Numerical Treatment of Elliptic Systems on Manifolds , 2001, Adv. Comput. Math..

[42]  W. Richard Bowen,et al.  Finite difference solution of the 2-dimensional Poisson–Boltzmann equation for spheres in confined geometries , 2002 .

[43]  I-Liang Chern,et al.  Accurate Evaluation of Electrostatics for Macromolecules in Solution , 2003 .

[44]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[45]  Pavel Dyshlovenko,et al.  Adaptive mesh enrichment for the Poisson-Boltzmann equation , 2001 .

[46]  B. Honig,et al.  A rapid finite difference algorithm, utilizing successive over‐relaxation to solve the Poisson–Boltzmann equation , 1991 .

[47]  Weihua Geng,et al.  Treatment of charge singularities in implicit solvent models. , 2007, The Journal of chemical physics.

[48]  Frédéric Gibou,et al.  An efficient fluid-solid coupling algorithm for single-phase flows , 2009, J. Comput. Phys..

[49]  Michael J. Holst,et al.  Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems , 2000, J. Comput. Chem..