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

Departement Wiskunde, University of Leuven, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
School of Physics, Huazhong University of Science and Technology, Wuhan 430007, China
Author to whom correspondence should be addressed.
Plasma 2018, 1(2), 242-258;
Received: 14 September 2018 / Revised: 4 October 2018 / Accepted: 5 October 2018 / Published: 9 October 2018
(This article belongs to the Special Issue Multiscale Methods in Plasma Physics)
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. View Full-Text
Keywords: particle-in-cell; Poisson–Boltzmann; multiscale material modeling particle-in-cell; Poisson–Boltzmann; multiscale material modeling
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Lapenta, G.; Jiang, W. Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation. Plasma 2018, 1, 242-258.

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