As strictly hyperbolic system of conservation laws of the form \[ u_{t}+f(u)_x =0 , \quad u(0,x)=\bar u (x) \] is considered, where \( u \in\mathbb{R}^N\), \(f:\mathbb{R}^N \rightarrow\mathbb{R}^N\) is smooth, especially from a numerical point of view, that means, a semidiscrete upwind scheme of this equation is investigated. If we suppose that the initial data \(\bar u (x) \) of this problem have small total variation the author proves that the solution of the upwind scheme \[ {\partial u(t,x) \over \partial t} + { ( f(u(t,x))-f(u(t,x-\varepsilon))) \over \varepsilon} =0 \] has uniformly bounded variation (BV) norm independent on \(t\) and \(\varepsilon\). Moreover the Lipschitz-continuous dependence of the solution of the upwind scheme \(u^{\varepsilon}(t)\) on the initial data is proved. This solution \(u^{\varepsilon}(t)\) converges in \( L_1\) to a weak solution of the corresponding hyperbolic system as \( \varepsilon \rightarrow 0\). This weak solution coincides with the trajectory of a Riemann semigroup which is uniquely determined by the extension of Liu's Riemann solver to general hyperbolic systems.