Discontinuous finite elements and godunov-type methods for the eulerian equations of gas dynamics

We investigate the numerical solution of typical problems in gas dynamics by discontinuous finite element methods, and compare the results with computations performed with variants of Godunov's conservative method.A one-dimensional shock wave problem with reflection, and a three-dimensional shock tube-type problem with convergent-divergent nozzle geometry are analyzed. For the one-dimensional problem we also present results obtained with a variant of Glimm's method. In one dimension, finite elements give valuable results, although they need a substantially larger computing time; in three space dimensions discontinuous elements appear to be too cumbersome, in the present form, to lend themselves to an efficient treatment of time-dependent shock wave problems.