High resolution schemes for the Euler equation are studied to calculate shock-capturing and propagating wave problems. In general, monotone methods, having total variation diminishing(TVD) property and satisfing the entropy condition, are used to remove oscillatory results in the case of upwind schemes. However, they are at most first order accurate. For the higher order accuracy, two types of schemes are employed and compared. First, limiter methods based on the second order MUSCL(monotonic upstream centered scheme for conservation laws) are used, which are TVD,and easy to extend to the multi-dimensions and converge fast. Secondly ENO(essentially nonoscillation ) method are used, which can be extended to higher order scheme and are not necessarilly TVD. The modified Roe scheme is used as a basic scheme for the both types of schemes. An aritificial compression method(ACM), improving resolution of the contact discontinuity, is applied for the both type of schemes, even though the ACM method was introduced for the ENO schemes. Two types of schemes are tested for the simple problems of the scalar wave, shock tube and the blast wave. More complicated problems are also solved for the steady flow around an arc bump and for the unsteady flow of moving schock in the presonce of a cylinder. It turns out the Minmod-ACM, which is in the first type is fast in convergence and reasonably accurate coparing with the ENO-ACM. Finally, a moving shock through a tube with abruptly varing area is calculated with the Minmod-ACM. Reflected waves after passing the larger area are propagated toward the inlet for the case of subsonic flow behind the shock, but the reflected waves are accumulated at the starting position of larger area for the case of supersonic flow behind the schock.
