1 research outputs found
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 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. 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
number of atoms in the protein studies, and is over 20 times faster than the
conditionally stable ADI methods