A Method of Boundary Equations for Nonlinear Poisson–Boltzmann Equation Arising in Biomolecular Systems

In this paper, we present a robust and accurate numerical algorithm for solving the nonlinear Poisson–Boltzmann equation, based on the difference potentials method (DPM). First, we remove singularity by a simple regularization technique and employ Newton’s method for linearizing the equation. Then we use the difference potentials method combined with a second-order finite difference scheme and curvilinear approach to solve the problem in the regions with a smooth general-shaped interface. Unlike many other methods, DPM does not need to treat the nonhomogeneous interface conditions resulting from the regularization and can handle discontinuity in the interface without loss of accuracy. In DPM the grid does not need to conform to the boundary or interfaces. This method approximates the resulting equation on a regular structured grid, which entails a low computational complexity, and does not face the challenge of reducing the order of accuracy near the nonconforming interfaces. Several numerical experiments are presented for illustrating the efficiency and accuracy of the developed numerical method.