A number of high-order methods are tested on a variety of dynamic problems in one, two, and three space dimensions. These problems include wave propagation phenomena as well as an asymptotic approach to a steady state. Both smooth and shocked flows are considered. The methods compared require only minor modifications of many existing second-order schemes. The results show that significant gains can be expected from the use of fourth-order methods. Spectral methods are also considered for some of the problems presented.