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

Implicit temporal discretization and exact energy conservation for particle methods applied to the Poisson-Boltzmann equation