  This work proposes a fast iterative method for local steric
Poisson--Boltzmann (PB) theories, in which the electrostatic potential is
governed by the Poisson's equation and ionic concentrations satisfy equilibrium
conditions. To present the method, we focus on a local steric PB theory derived
from a lattice-gas model, as an example. The advantages of the proposed method
in efficiency are achieved by treating ionic concentrations as scalar implicit
functions of the electrostatic potential, though such functions are only
numerically achievable. The existence, uniqueness, boundness, and smoothness of
such functions are rigorously established. A Newton iteration method with
truncation is proposed to solve a nonlinear system discretized from the
generalized PB equations. The existence and uniqueness of the solution to the
discretized nonlinear system are established by showing that it is a unique
minimizer of a constructed convex energy. Thanks to the boundness of ionic
concentrations, truncation bounds for the potential are obtained by using the
extremum principle. The truncation step in iterations is shown to be energy and
error decreasing. To further speed-up computations, we propose a novel
precomputing-interpolation strategy, which is applicable to other local steric
PB theories and makes the proposed methods for solving steric PB theories as
efficient as for solving the classical PB theory. Analysis on the Newton
iteration method with truncation shows local quadratic convergence for the
proposed numerical methods. Applications to realistic biomolecular solvation
systems reveal that counterions with steric hindrance stratify in an order
prescribed by the parameter of ionic valence-to-volume ratio. Finally, we
remark that the proposed iterative methods for local steric PB theories can be
readily incorporated in well-known classical PB solvers.
