  Wereportonanewmultiscalemethodapproachforthestudyofsystemswith wide
separation of short-range forces acting on short time scales and long-range
forces acting on much slower scales. We consider the case of the
Poisson-Boltzmann equation that describes the long-range forces using the
Boltzmann formula (i.e. we assume the medium to be in quasi local thermal
equilibrium). We developed a new approach where fields and particle information
(mediated by the equations for their moments) are solved self-consistently. The
new approach is implicit and numerically stable, providing exact energy
conservation. We tested different implementations all leading to exact energy
conservation. The new method requires the solution of a large set of non-linear
equations. We considered three solution strategies: Jacobian Free Newton
Krylov, an alternative, called field hiding, based on hiding part of the
residual calculation and replacing them with direct solutions and a Direct
Newton Schwarz solver that considers simplified single particle-based Jacobian.
The field hiding strategy proves to be the most efficient approach.
