Commun. Comput. Phys., 12 (2012), pp. 1-41.


High Order Schemes on Three-Dimensional General Polyhedral Meshes - Application to the Level Set Method

Thibault Pringuey 1*, R. Stewart Cant 1

1 CFD Laboratory, Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK.

Received 26 May 2011; Accepted (in revised version) 5 August 2011
Available online 16 January 2012
doi:10.4208/cicp.260511.050811a

Abstract

In this article, we detail the methodology developed to construct arbitrarily high order schemes - linear and WENO - on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

AMS subject classifications: 65M08, 76-04, 76N99

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Key words: WENO scheme, three-dimensional, unstructured mesh, mixed element, polyhedral element, hyperbolic equations, level set.

*Corresponding author.
Email: tp299@cam.ac.uk (T. Pringuey), rsc10@cam.ac.uk (R. S. Cant)
 

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