(BQ) Part 2 book "The finite element method" has contents: Compressible high-speed gas flow, shallow-water problems, waves, computer implementation of the CBS algorithm. Invite you to reference. | 6 Compressible high-speed gas ¯ow Introduction Problems posed by high-speed gas ¯ow are of obvious practical importance. Applications range from the exterior ¯ows associated with ¯ight to interior ¯ows typical of turbomachinery. As the cost of physical experiments is high, the possibilities of computations were explored early and the development concentrated on the use of ®nite di erence and associated ®nite volume methods. It was only in the 1980s that the potential o ered by the ®nite element forms were realized and the ®eld is expanding rapidly. One of the main advantages in the use of the ®nite element approximation here is its capability of ®tting complex forms and permitting local re®nement where required. However, the improved approximation is also of substantial importance as practical problems will often involve three-dimensional discretization with the number of degrees of freedom much larger than those encountered in typical structural problems (105 ±107 DOF are here quite typical). For such large problems direct solution methods are obviously not practicable and iterative methods based generally on transient computation forms are invariably used. Here of course we follow and accept much that has been established by the ®nite di erence applications but generally will lose some computational e ciency associated with structured meshes typically used here. However, the reduction of the problem size which, as we shall see, can be obtained by local re®nement and adaptivity will more than compensate for this loss (though of course structured meshes are included in the ®nite element forms). In Chapters 1 and 3 we have introduced the basic equations governing the ¯ow of compressible gases as well as of incompressible ¯uids. Indeed in the latter, as in Chapter 4, we can introduce small amounts of compressibility into the procedures developed there speci®cally for incompressible ¯ow. Here we shall deal with highspeed ¯ows with Mach numbers generally in excess
