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

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.