  This work further improves the pseudo-transient approach for the Poisson
Boltzmann equation (PBE) in the electrostatic analysis of solvated
biomolecules. The numerical solution of the nonlinear PBE is known to involve
many difficulties, such as exponential nonlinear term, strong singularity by
the source terms, and complex dielectric interface. Recently, a pseudo-time
ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao,
Applied Mathematics and Computation, 380, 125267, (2020)], by analytically
handling both nonlinearity and singular sources. The GFM interface treatment
not only captures the discontinuity in the regularized potential and its flux
across the molecular surface, but also guarantees the stability and efficiency
of the time integration. However, the molecular surface definition based on the
MSMS package is known to induce instability in some cases, and a nontrivial
Lagrangian-to-Eulerian conversion is indispensable for the GFM finite
difference discretization. In this paper, an Eulerian Solvent Excluded Surface
(ESES) is implemented to replace the MSMS for defining the dielectric
interface. The electrostatic analysis shows that the ESES free energy is more
accurate than that of the MSMS, while being free of instability issues.
Moreover, this work explores, for the first time in the PBE literature,
adaptive time integration techniques for the pseudo-transient simulations. A
major finding is that the time increment $\Delta t$ should become smaller as
the time increases, in order to maintain the temporal accuracy. This is
opposite to the common practice for the steady state convergence, and is
believed to be due to the PBE nonlinearity and its time splitting treatment.
Effective adaptive schemes have been constructed so that the pseudo-time GFM
methods become more efficient than the constant $\Delta t$ ones.
