An original 2nd-order Method of Characteristics (MOC) which it works with a five-point interpolating scheme valid for solving the hyperbolic and quasilinear partial differential equations that describe the transient flow phenomenon in pipelines is shown. The results obtained by both MOC 2nd-order and exact solution (MOC 1st-order) are compared. It is shown that MOC 2nd-order allows obtain near-to-exact results within a wide range of Courant numbers.