Introduction#
In the TRUST code, different numerical schemes are availabe to the user : VDF, VEF and the PolyMAC family.
The VDF discretisation is based on the Marker and Cell scheme presented in [Harlow et al., 1965].
The VEF discretisation is based on the Crouzeix-Raviart element method [Emonot, 1992].
The PolyMAC discretisation family has been developped since 2018. Three PolyMAC are usable in TRUST. They have been built using a Finite Volume (FV) framework on a staggered mesh so as to extend the MAC scheme developped in [Harlow et al., 1965] to complex grids:
PolyMAC : based on a Compact Discrete Operator (CDO) approach, such as the one presented in [Bonelle, 2014] and [Milani, 2020].
PolyMACP0 : based on MPFA approach, such as the one presented in [Agelas and Masson, 2008], [Droniou, 2014] and [Le Potier, 2017].
PolyMACP0P1NC : based on a Hybrid Finite Volmue (HFV) approach, such as the one presented in [Eymard et al., 2007] and [Droniou et al., 2010].
Thereafter, for each method the core ideas and the main steps for the discretisation of the incompressible Navier-Stokes equation are presented. For now, the PolyMAC and PolyMAC_P0 parts are completed, the others are a work in progress.
Notations#
Let’s consider a space \(\Omega\) and a certain grid \(\mathcal{M}\) of non-overlapping polyhedrons that map \(\Omega\).
In the following:
A polyhedron of the grid will be called a cell : \(e\).
A face \(f\) is defined as the intersection of two cells or one face and a boundary. Faces of the grid are supposed to be planar.
An edge \(\sigma\) is defined as the intersection of faces or faces and boundary. This entity only exists in the 3D framework.
A vertex \(v\) is defined as the intersection of edges or edges and a boundary.
The set of cells will be called \(E\). The set of faces of a peculiar cell \(e\) will be denoted \(F_e\). In the same fashion, the set of edges of a peculiar face \(f\) will be noted as \(\Sigma _f\) and finally, the two vertices of an edge \(\sigma\) will be denoted \(V_{\sigma}\). In the rest of the document, the measure of an unknown \(x\) at a control volume \(cv\) will be denoted:
where \(|\cdot|\) will be a global measure operator over the considered control volume. For example, \(|e|\) refers to the volume of the cell \(e\), \(|f|\) to the surface of the face \(f\) and \(|\sigma|\) to the length of the edge \(\sigma\). Unknown \(u\) refers to the velocity and \(p\) refers to the pressure.
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