A Genuinely Multidimensional Upwind Scheme and an Efficient Multigrid for the Compressible Euler Equations

We present a new approach towards the construction of a genuinely multidimensional high-resolution scheme for computing steady-state solutions of the Euler equations of gas dynamics. The unique advantage of this approach is that the Gauss-Seidel relaxation is stable when applied directly to the high-resolution discrete equations, thus allowing to obtain a very good e ciency of the multigrid steady-state solver. This is the only high-resolution scheme known to us that has this property. The two-dimensional scheme is presented in details. It is formulated on triangular (structured and unstructured) meshes and can be interpreted as a genuinely two-dimensional extension of the Roe scheme. The quality of the solutions obtained using this scheme and the performance of the multigrid algorithm are illustrated by the numerical experiments. Construction of the threedimensional scheme is outlined brie y as well. This research was supported by DOE grant DE-FG02-92ER25139 while the author was in residence at Courant Institute of Mathematical Sciences, 251 Mercer st., New York University, New York, NY 10012. It was also supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering, (ICASE), NASA Langley Research Center, Hampton, VA 23681. i