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• Overview of discretisation
• 1D order of accuracy and cubicFit/highOrderFit matrix comparison

Implementation plan

1. Make stencil boundary value inclusion/exclusion configurable ✅
2. Calculate least squares pseudo-inverse without multipliers ✅
3. Calculate least squares pseudo-inverse with multipliers ✅
4. Calculate vertices of cell faces ✅
5. mapDistribute stencil cellVertices across processors ✅
6. Translate stencil with target face centre as coordinate origin ✅
7. Rotate stencil with Sf as x direction ✅
8. Hard-code volume moments that are calculated by hand for each cell of a uniform mesh with 1×1×1 cells. At this point I can perform convergence tests on uniform meshes. ✅
9. Generate collection of tetrahedra from cellVertices
10. Calculate volume moments for tetrahedra
11. Decompose target face into triangles
12. Translate and rotate target face into local coordinates
13. Calculate surface moments for triangles
14. Calculate weights using volume moments and surface moments

Test cases

• solid body rotation in 2D on uniform and distorted meshes following Leonard et al. 1996, Chen et al. 2017
• horizontal flow above a mountain on a basic terrain-following mesh (Schär et al. 2002, Shaw et al. 2017)

Optionally

• deformational flow in 2D (Lauritzen et al. 2012; Chen et al. 2017) will need periodic BCs
• deformational flow on a spherical Earth following Lauritzen et al. 2012, although special high-order geometry might be necessary on the sphere (Sjögreen 2012)
• 3D solid body rotation (Smolarkiewicz and Szmelter 2005)