TY - JOUR
T1 - Unconditionally Stable Time Splitting Methods for the Electrostatic Analysis of Solvated Biomolecules
AU - L. Wilson & S. Zhao
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 852
EP - 878
PY - 2016
DA - 2016/11
SN - 13
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/469.html
KW - 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.
AB - <p style="text-align: justify;">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,
the nonlinear term is analytically integrated, so that the difficulties in direct treatment of the
strong nonlinearity can be bypassed. However, in a pseudo-time NPB computation, the use of
large time increments is necessary to reach the steady state efficiently. The existing alternating
direction implicit (ADI) methods for the transient 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 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. By using
finite differences in space and implicit integrations in time, the numerical orders of the proposed
schemes are found to be one in both space and time. 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 in treating 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 matrix in each one dimensional subsystem,
the overall computation is very efficient. The unconditionally stable LOD method scales linearly
with respect to the number of atoms in the protein studies, and is over 20 times faster than the
conditionally stable ADI methods.</p>