  Recently, a nonlinear Poisson equation has been introduced to model nonlinear
and nonlocal hyperpolarization effects in electrostatic solute-solvent
interaction for biomolecular solvation analysis. Due to a strong nonlinearity
associated with the heterogeneous dielectric media, this Poisson model is
difficult to solve numerically, particularly for large protein systems. A new
pseudo-transient continuation approach is proposed in this paper to efficiently
and stably solve the nonlinear Poisson equation. A Douglas type alternating
direction implicit (ADI) method is developed for solving the pseudo-time
dependent Poisson equation. Different approximations to the dielectric profile
in heterogeneous media are considered in the standard finite difference
discretization. The proposed ADI scheme is validated by considering benchmark
examples with exact solutions and by solvation analysis of real biomolecules
with various sizes. Numerical results are in good agreement with the
theoretical prediction, experimental measurements, and those obtained from the
boundary value problem approach. Since the time stability of the proposed ADI
scheme can be maintained even using very large time increments, it is efficient
for electrostatic analysis involving hyperpolarization effects.
