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

This work introduces novel unconditionally stable operator splitting methods for solving the time dependent nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic analysis of solvated biomolecules. In a pseudo-transient continuation solution of the NPB equation, a long time integration is needed to reach the steady state. This calls for time stepping schemes that are stable and accurate for large time increments. The existing alternating direction implicit (ADI) methods for the NPB equation are known to be conditionally stable, although being fully implicit. To overcome this difficulty, we propose several new operator splitting schemes, in both multiplicative and additive styles, including locally one-dimensional (LOD) schemes and additive operator splitting (AOS) schemes. The nonlinear term is integrated analytically in these schemes, while standard discretizations with finite differences in space and implicit time integrations are used. The proposed schemes become much more stable than the ADI methods, and some of them are indeed unconditionally stable in dealing with solvated proteins with source singularities and non-smooth solutions. Numerically, the orders of convergence in both space and time are found to be one. Nevertheless, the precision in calculating the electrostatic free energy is low, unless a small time increment is used. Further accuracy improvements are thus considered, through constructing a Richardson extrapolation procedure and a tailored recovery scheme that replaces the fast Fourier transform method by the operator splitting method in the vacuum case. After acceleration, the optimized LOD method can produce a reliable energy estimate by integrating for a small and fixed number of time steps. Since one only needs to solve a tridiagonal linear system in each independent one dimensional process, the overall computation is very efficient. The unconditionally stable LOD method scales linearly with respect to the ∗Corresponding author. Tel: 1-205-3485303, Fax: 1-205-3487067, Email: [email protected] 1 ar X iv :1 41 0. 27 88 v1 [ m at h. N A ] 1 0 O ct 2 01 4 number of atoms in the protein studies, and is over 20 times faster than the conditionally stable ADI methods. Keyword: Nonlinear Poisson-Boltzmann equation; Pseudo-transient continuation approach; Time splitting; Alternating direction implicit (ADI); Locally one dimensional (LOD); Additive operator splitting (AOS); Electrostatic free energy. MSC: 65M06, 92-08, 92C40,