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Difference schemes for linear hyperbolic systems are considered. As a main result, a weak derivative form (WDF) of the governing equations is derived, which is also valid near flow discontinuities. The occurrence of one-sided derivatives in the WDF structure indicated how to difference near discontinuities. When first-order differencing is applied to the WDF result, the (linearly identical) schemes by Godunov, Roe, and Steger-Warming are reproduced. The extension to nonlinear systems is via a local linearization. Choosing Roe's averaging reduces the WDF algorithm to Roe's scheme, whereas other nonlinear WDF schemes are possible. The suitability of various kinds of averaging is numerically investigated. For weak shocks a surprising lack of sensitivity of the method to a particular averaging is exhibited. However, for strong shocks and where the ordinary arithmetic average is used, a slightly more pronounced difference in performance exists between Roe's scheme and WDF.