Summary: The exponential time differencing method (ETD) has been recently introduced for gas dynamics problems, through which significant speedup against the popular TVD Runge-Kutta scheme can be gained. In this paper, a local ETD time advancing method is introduced in solving fluid dynamics problems with smooth solutions, which has the natural characteristic of parallelism. The application of four ETD schemes in conjunction with weighted essentially non-oscillation spatial discrete scheme is discussed, in which case the nearly analytic Jacobian matrix is obtained using the chain rule, and the matrix exponential computation is approximated through Krylov subspace projection method. For large scale simulations, the local version of four ETD schemes are introduced on the basis of overlapping domain decomposition method. Addictive Schwarz iteration is applied to decouple the multi-domain problem, and stationary problems are solved on the sub-domains in every time step. Numerical tests against several one- and two-dimensional problems demonstrate that the present method is accurate and efficient. Schwarz iteration does not increase the computation load much, and the numerical error of local ETD method will converge when overlap size is large enough.