A distinguishing feature of upwind numerical methods is this: the discretisation of the equations on a mesh is performed according to the direction of propagation of information on that mesh. In this way, salient features of the physical phenomena modelled by the equations are incorporated into the discretisation schemes. There are essentially two approaches for identifying upwind directions, namely the Godunov approach [216] studied in Chap. 6, and the Flux Vector Splitting (FVS) approach [424], [463], [560], [561] to be studied in this chapter. These two approaches are often referred to as the Riemann approach and the Boltzmann approach [244]. The respective numerical methods derived from these two approaches are often referred to as Flux Difference Splitting Methods and Flux Vector Splitting Methods.