These html pages are based on the PhD thesis "Cluster-Based Parallelization of Simulations on Dynamically Adaptive Grids and Dynamic Resource Management" by Martin Schreiber.
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2.9 Rotational invariancy and edge space

Working with triangles, a straightforward evaluation of the term F(Ui(t)) ⋅⃗n(x,y) involving both flux functions G(U) and H(U) followed by a multiplication with the outward pointing normal can be optimized. We use the so-called rotational invariancy [Tor01], with flux functions for the hyperbolic systems considered in this thesis holding a crucial property.

We consider a two-dimensional normal vector ⃗ne = (cos(α),sin(α))T pointing outward the edge e. Then, for the computation of the flux update, the equation

F (U) ⋅⃗n = G(U )⋅n + H (U) ⋅n = R(α)-1F (R(α)(U )) e x y
(2.13)

holds true for a given rotation matrix R(α). The matrix is setup with an n-dimensional rotation matrix with the entries stored to the n×n direction dependent components such as the velocity and momentums and 1 on the diagonal for direction independent components. This matrix rotates, e.g., orientation-dependent components such as the velocity and momentum which are parallel to ⃗ne to the x-axis (see Appendix A.1.7 for an example).

For later purpose, the right hand side of Equation (2.13) is further described:

1.
To compute the flux across an edge, the conserved quantities depending on a direction are first rotated from the reference space to the so-called edge space [AS98].
2.
Then, the change of conserved quantities is evaluated in the one-dimensional edge space.
3.
Finally, the updates related to the flux are rotated back to reference space and applied to the conserved quantities stored in the reference space.

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