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This work introduces novel unconditionally stable operator splitting methodsfor solving the time dependent nonlinear Poisson-Boltzmann (NPB) equation forthe electrostatic analysis of solvated biomolecules. In a pseudo-transientcontinuation solution of the NPB equation, a long time integration is needed toreach the steady state. This calls for time stepping schemes that are stableand accurate for large time increments. The existing alternating directionimplicit (ADI) methods for the NPB equation are known to be conditionallystable, although being fully implicit. To overcome this difficulty, we proposeseveral new operator splitting schemes, in both multiplicative and additivestyles, including locally one-dimensional (LOD) schemes and additive operatorsplitting (AOS) schemes. The proposed schemes become much more stable than theADI methods, and some of them are indeed unconditionally stable in dealing withsolvated proteins with source singularities and non-smooth solutions.Numerically, the orders of convergence in both space and time are found to beone. Nevertheless, the precision in calculating the electrostatic free energyis low, unless a small time increment is used. Further accuracy improvementsare thus considered. After acceleration, the optimized LOD method can produce areliable energy estimate by integrating for a small and fixed number of timesteps. Since one only needs to solve a tridiagonal linear system in eachindependent one dimensional process, the overall computation is very efficient.The unconditionally stable LOD method scales linearly with respect to thenumber of atoms in the protein studies, and is over 20 times faster than theconditionally stable ADI methods.

Unconditionally stable time splitting methods for the electrostatic analysis of solvated biomolecules