A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations

A fast finite difference method based on the monotone iterative method and the fast Poisson solver on irregular domains for a 2D nonlinear Poisson–Boltzmann equation is proposed and analyzed in this paper. Each iteration of the monotone method involves the solution of a linear equation in an exterior domain with an arbitrary interior boundary. A fast immersed interface method for generalized Helmholtz equations on exterior irregular domains is used to solve the linear equation. The monotone iterative method leads to a sequence which converges monotonically from either above or below to a unique solution of the problem. This monotone convergence guarantees the existence and uniqueness of a solution as well as the convergence of the finite difference solution to the continuous solution. A comparison of the numerical results against the exact solution in an example indicates that our method is second order accurate. We also compare our results with available data in the literature to validate the numerical method. Our method is efficient in terms of accuracy, speed, and flexibility in dealing with the geometry of the domain

[1]  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..

[2]  Xin Lu,et al.  Block Monotone Iterations for Numerical Solutions of Fourth-Order Nonlinear Elliptic Boundary Value Problems , 2003, SIAM J. Sci. Comput..

[3]  Seth Fraden,et al.  Solving the Poisson–Boltzmann equation to obtain interaction energies between confined, like-charged cylinders , 1998 .

[4]  Huajian Gao,et al.  A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid , 1999 .

[5]  N. E. Hoskin.  The interaction of two identical spherical colloidal particles - The interaction of two identical spherical colloidal particles. I. Potential distribution , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[6]  P. Swarztrauber,et al.  Efficient Fortran subprograms for the solution of separable elliptic partial differential equations , 1979 .

[7]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[8]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[9]  Roland A. Sweet,et al.  Algorithm 541: Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations [D3] , 1979, TOMS.

[10]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[11]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[12]  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 .

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

[14]  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..

[15]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

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

[17]  Adel O. Sharif,et al.  Long-range electrostatic attraction between like-charge spheres in a charged pore , 1998, Nature.

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

[19]  Daniel Harries.  Solving the Poisson−Boltzmann Equation for Two Parallel Cylinders , 1998 .

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

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

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

[23]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[24]  Derek Y. C. Chan,et al.  Computation of forces between spherical colloidal particles : nonlinear Poisson-Boltzmann theory , 1994 .

[25]  C. V. Pao,et al.  Block monotone iterative methods for numerical solutions of nonlinear elliptic equations , 2022 .

[26]  Stanley Osher,et al.  A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow , 1997 .

[27]  C. V. Pao,et al.  Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion , 1989 .

[28]  Zhilin Li.  The immersed interface method: a numerical approach for partial differential equations with interfaces , 1995 .

[29]  I. Lakatos,et al.  Colloids Surfaces A: Physicochem , 1998 .

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

[31]  C. V. Pao,et al.  Monotone iterative methods for finite difference system of reaction-diffusion equations , 1985 .