Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation

Giovanni Lapenta,Wei Jiang

doi:10.3390/plasma1020021

Giovanni Lapenta, Wei Jiang

Open Access

https://doi.org/10.3390/plasma1020021

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Journal: Plasma	Publication Date: Oct 9, 2018
Citations: 5	License type: CC BY 4.0

Affiliation: KU Leuven, Huazhong University of Science and Technology

Abstract

We report on a new multiscale method approach for the study of systems with 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 develop 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 test different implementations that all lead to exact energy conservation. The new method requires the solution of a large set of non-linear equations. We consider three solution strategies: Jacobian Free Newton Krylov, an alternative, called field hiding which is based on hiding part of the residual calculation and replacing them with direct solutions and a Direct Newton Schwarz solver that considers a simplified, single, particle-based Jacobian. The field hiding strategy proves to be the most efficient approach.

Highlights

Matter in any form or state is characterized by the presence of particles interacting via fields.At the most fundamental level, a quantum point of view is needed, but as larger and larger portions of matter need to be studied, attention shifts from the quantum level to the particle level, to the macroscopic level.Perhaps the key effort in science, and in particular, in computational science, is to design models that are able to describe or predict the properties and behavior of matter based on the knowledge of its constituents and their interactions: We consider here in particular the particle approach [1,2]
To test the energy conservation and the computational implementation of the Jacobian–Free Newton Krylov (JFNK) methods, we report the result of one sample problem
The results presented are converged in the sense that the location of the features in the phase space does not change

Summary

Introduction

Matter in any form or state is characterized by the presence of particles interacting via fields. Perhaps the key effort in science, and in particular, in computational science, is to design models that are able to describe or predict the properties and behavior of matter based on the knowledge of its constituents and their interactions: We consider here in particular the particle approach [1,2]. Particle models must account for the presence of long and short-range interactions that act on different temporal scales. The contributions from short-range forces can be computed for each particle considering only the others within a short distance. The long-range forces require a global approach for the whole system. One class of Plasma 2018, 1, 242–258; doi:10.3390/plasma1020021 www.mdpi.com/journal/plasma

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