On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

• Published in: International Journal of Differential Equations
• Main Authors: Helgadottir, Asdis, Guittet, Arthur, Gibou, Frédéric
• Other Authors: Iðnaðarverkfræði-, vélaverkfræði- og tölvunarfræðideild (HÍ), Faculty of Industrial Eng., Mechanical Eng. and Computer Science (UI), Verkfræði- og náttúruvísindasvið (HÍ), School of Engineering and Natural Sciences (UI), Háskóli Íslands, University of Iceland
• Format: Article in Journal/Newspaper
• Language: English
• Published: Hindawi Limited 2018
• Subjects: Töluleg greining; Stærðfræðileg tölfræði; Iceland
• Online Access: https://hdl.handle.net/20.500.11815/1086; https://doi.org/10.1155/2018/9216703

Publisher's version (útgefin grein) We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the -norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand. The research of Á. Helgadóttir was supported by the University of Iceland Research Fund 2015 under HI14090070. The researches of A. Guittet and F. Gibou were supported in part by the NSF under DMS-1412695 and DMREF-1534264. Peer Reviewed