Having discussed quantum transport models in the preceding two chapters, we now turn to the semi-classical Boltzmann descriptions and their moment equations in the hydrodynamic model in electron transport in complex media including semiconductors and plasmas. Then, finite difference methods for solving the hydrodynamic equations for semiconductor devices will be discussed. Because of the high field effect in sub-micron devices, the electron velocity may develop a sharp transition profile resembling shock waves [1], as in high-speed gas dynamics. Therefore, shock capturing schemes developed for gas dynamics [2, 3] can be applied for device simulations. Here, we will present three methods: the traditional Godunov methods, the weighted essentially non-oscillatory (ENO) finite difference methods, and the central differencing methods. It should be mentioned that the discontinuous Galerkin method can also be used to compute the semiconductor hydrodynamic equations [4].