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

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Abstract

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

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Authors and Affiliations

  1. Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Zhilin Li

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    C. V. Pao

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong

    Zhonghua Qiao

Authors
  1. Zhilin Li
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  2. C. V. Pao
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  3. Zhonghua Qiao
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Correspondence to Zhilin Li.

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Li, Z., Pao, C.V. & Qiao, Z. A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations. J Sci Comput 30, 61–81 (2007). https://doi.org/10.1007/s10915-005-9019-y

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  • DOI: https://doi.org/10.1007/s10915-005-9019-y

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