SUMMARYIn this paper numerical methods for solving ÿrst-order hyperbolic partial di erential equations are developed. These methods are developed by approximating the ÿrst-order spatial derivative by third-order ÿnite-di erence approximations and a matrix exponential function by a third-order rational approximation having distinct real poles. Then parallel algorithms are developed and tested on a sequential computer for an advection equation with constant coe cient and a non-linear problem.