Multi-D upwinding and multigridding for steady Euler flow computations

Multi-dimensional upwind discretization. for the steady Euler equations are studied, with the emphasis on both a good accuracy and a good efficiency. The discretizations consist of a one-dimensional Riemann solver with locally rotated left and right cell face states, the rotation angle depending on the local flow solution. First, on the basis of a linear, scalar model equation, a study is made of the accuracy and stability properties of these schemes. Next the extension is made to the steady Euler equations. It is shown that for Euler flows, an appropriate local rotation angle can be found by maximizing a Riemann invariant along the middle subpath of the wave path in state space. For the steady, two-dimensional Euler equations, numerical results are presented for some supersonic test cases with either oblique contact discontinuity or oblique shock wave.