A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs et al., we construct an appropriate constrained free energy functional, such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. This is a formulation that searches for a true minimum in function space, in contrast to previous variational approaches that rather searched for a saddle point. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Some results are presented for a charged colloidal sphere surrounded by counterions, and optimizations based upon Fast Fourier Transforms and hierarchical pre-conditioning are briefly discussed.
The zeroth-order perturbation corresponds to the Poisson-Boltzmann equation, which is solved by a constrained variational approach suggested in REF .
Simple and Robust Solver for the Poisson-Boltzmann Equation
arXiv: Soft Condensed Matter
Physics Mathematics Medicine