The Spectral Difference Scheme

Some Theory...

Subsonic Inviscid Flow

Figure 1 shows a solution of the Euler equations for flow around the NACA0012 airfoil using the $ 3^{rd}$ order Spectral Difference scheme at flow conditions $ M=0.3$ and zero angle of attack. Mach number contours are shown on the left, while the figure on the right compares the entropy error with results from a finite volume scheme, which employs the CUSP scheme with a SLIP data reconstruction [7], and uses the triangles as control volumes.
Figure 1: Validation of the Spectral Difference Method: Subsonic flow around the NACA0012 airfoil.

Note that the entropy error for the spectral difference scheme with 15,360 DOF is roughly equal to the entropy error of the finite-volume scheme with approximately $ 40,000$ DOF

Transonic/Supersonic Inviscid Flow

Some Theory...

The famous mach reflection case...

The shocks are very well captured using a local-projection type limiter, very similar to the one proposed for Discontinuous Galerkin methods.

Viscous Flow

A validation for low-Reynolds-number flow can be seen below
Figure 2: Validation of the Spectral Difference Method: Subsonic viscous flow around the NACA0012 airfoil. Left: Spectral Difference, 3rd order. Right: Computational mesh

The 3rd order SD scheme resolves the flow even on this coarse mesh. This is not the case for a 2nd order finite volume scheme as is shown below, which clearly demonstrates the superiority of the higher order scheme.
Figure 2: Subsonic viscous flow around the NACA0012 airfoil. Finite Volume Schemes, 2nd order. Right: Computational mesh

Georg May 2006-02-23