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

TL;DR

This paper introduces a new implicit multiscale particle method for the Poisson-Boltzmann equation that conserves energy exactly and is numerically stable, effectively handling systems with disparate force and time scales.

Contribution

The paper presents a novel implicit approach with exact energy conservation for particle methods applied to the Poisson-Boltzmann equation, including efficient solution strategies.

Findings

The new method achieves exact energy conservation.

Field hiding is the most efficient solution strategy.

The approach is stable for multiscale systems.

Abstract

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.