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2.7 Flux matrices E


Figure 2.4: Sketch of flux evaluation. Right handed image: The conserved quantities UL and UR are in the triangles in world space evaluated at edge quadrature points. Left handed image: After projecting the conserved quantities to the edge space, the flux computation is then based on one-dimensional conserved quantities. These quantities are then used for the flux computation in edge space.

The flux term dT (F( iUi(t)φi(ξ,η))φj(ξ,η)) ⃗n(ξ,η) in Eq. (2.9) represents the change of conserved quantities via a flux across an edge. With the non-overlapping support of any two adjacent triangles, also the approximated solution is unsteady at the triangle boundaries. By considering each triangle edge separately, we can evaluate the edge-flux term in three steps:

First, we evaluate our approximated solution at particular quadrature points on the edge, see left image in Fig. 2.4. Here, we can use a beneficial property of the GL quadrature points (see Sec. 2.3.1): one-dimensional GL quadrature points on the edge directly coincide with the GL quadrature and are thus nodal points of the reference triangle. Therefore, computing the conserved quantities at the quadrature points only involves the conserved quantities stored at these points. See Appendix A.1.5 for the sparse matrix selecting the corresponding conserved quantities.
Second, the flux is computed at the quadrature points based on pairs of selected coinciding quadrature points from the local and the adjacent cell. We can simplify this two-dimensional Riemann problem to a one-dimensional one (see right image in Fig. 2.4) by changing the basis to the edge normal taken as the x-basis axis and the discontinuity on the edge at x = 0: Only considering a single conserved quantity on an edge, L and R represent the current solution on the left and right side of the y-axis, respectively. The change over time can then be computed by using flux solvers which is further discussed in Sec. 2.10.
Third, we reconstruct our approximated solution iF(Ui(t))φi(ξ,η) for each edge based on the computed flux crossing the edge [GW08]. By splitting the surface integral dT over the triangle reference boundaries into three separate integrals eE de over each edge e E (see Section 2.2) and factoring out the term with the computed flux approximation from the integral, this yields
    F◟(Ui(t)◝)◜⋅⃗n(ξ,η)◞ e φi(ξ,η )φj (ξ,η)
e∈E  Approximated flux
with the evaluation of the Riemann problem term F(Ui(t)) ⃗n(ξ,η) further described in Section 2.10. Note that the “Approximated flux” term may also depend on other conserved quantities than Ui. Solutions for this integral can again be stored in matrices similarly to the previous sections. For GL points, these matrices are again sparse and influence conserved quantities only if these are stored on the corresponding triangle’s edge (see Appendix A.1.5).

The presented method is a quadrature of the flux on each edge with fluxes evaluated at pairs of given quadrature points. Rather than evaluating the flux for several pairs of given points, alternative approaches use flux computations based on a single pair of given functions [AS98]. These functions represent the conserved quantities on the entire edge. Since the interfaces derived in this framework can be also used for such an implementation, we continue using the previously described method without loss of applicability.