Below are two choices in how to project interior values to edges: area averaging and simple averaging for an RT3 element with six interior points. The interior points in the example are provided sequential values and the edges are averaged from the neighboring interior values using either the included area of that interior node's triangulation edge span or a simple average.
| Area Averaged | Simple Average |
|---|---|
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Early experiments with the supersonic Wedge using this edge projection for only cells that have shocks in them shows a reduction in stability when using area averaging and even worse stability using simple averaging. The excursions move to the interior nodes in the vicinity of shock waves.
To figure out whether the averaging itself was problematic, I enabled the same projection for all cells, not just shocked cells. I obtained stable results for P=1, but in analyzing the unstable results for P=2 it is clear that instabilities start in the cell interiors near the walls and shock waves, but are more pronounced near the walls. There are also small instabilities present in the P=1 case near the walls, and so we can conclude that the suppression of the interior polynomial modes in the edge projections seems to be producing a compensating reaction in the interior polynomials of the cells.
On inspection of these interior nodes adjacent to the edge nodes, we can see that we have a nicely associated set of P points from the interior that are adjacent to the P+1 points on each edge. We can construct a P-1 polynomial along each edge using an orthogonal spatial projection of the interior node values that will be nicely spaced along each edge. The solution in the interior is of order P-1 (the RT element is one order higher), so this is actually coincident with the notion of extending solution values at their correct order to the boundary. We can think of it as extending a trace of the 2D polynomial solution to the boundary, almost like a camera shot of the interior projected to the edge.
My concern with this approach is that we will still expect some of the Gibbs phenomenon in the projection because the P-1 1D polynomial will encompass the large variation due to the shock. I do expect the severity of the excursions to be much reduced compared to the 2D extrapolation of values from the full interior. As such, this seems like a valid way to preserve the polynomial nature of the interior while reducing the severity of the excursions.