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This work proposes a fast iterative method for local stericPoisson--Boltzmann (PB) theories, in which the electrostatic potential isgoverned by the Poisson's equation and ionic concentrations satisfy equilibriumconditions. To present the method, we focus on a local steric PB theory derivedfrom a lattice-gas model, as an example. The advantages of the proposed methodin efficiency are achieved by treating ionic concentrations as scalar implicitfunctions of the electrostatic potential, though such functions are onlynumerically achievable. The existence, uniqueness, boundness, and smoothness ofsuch functions are rigorously established. A Newton iteration method withtruncation is proposed to solve a nonlinear system discretized from thegeneralized PB equations. The existence and uniqueness of the solution to thediscretized nonlinear system are established by showing that it is a uniqueminimizer of a constructed convex energy. Thanks to the boundness of ionicconcentrations, truncation bounds for the potential are obtained by using theextremum principle. The truncation step in iterations is shown to be energy anderror decreasing. To further speed-up computations, we propose a novelprecomputing-interpolation strategy, which is applicable to other local stericPB theories and makes the proposed methods for solving steric PB theories asefficient as for solving the classical PB theory. Analysis on the Newtoniteration method with truncation shows local quadratic convergence for theproposed numerical methods. Applications to realistic biomolecular solvationsystems reveal that counterions with steric hindrance stratify in an orderprescribed by the parameter of ionic valence-to-volume ratio. Finally, weremark that the proposed iterative methods for local steric PB theories can bereadily incorporated in well-known classical PB solvers.

Fast Newton Iterative Method for Local Steric Poisson--Boltzmann Theories in Biomolecular Solvation