In this paper, numerical boundary stabilisation of a non-uniform hyperbolic system of balance laws is studied. For the numerical discretisation of the balance laws, a first order explicit upwind scheme is used for the spatial discretisation; and for the temporal discretisation a splitting technique is employed. A discrete -Lyapunov function is employed to investigate conditions for the stability of the system. After constructing discrete numerical Lyapunov functionals, we prove an asymptotic exponential stability result for a class of non-uniform linear hyperbolic systems of balance laws. Convergence of the solution to its equilibrium is proved. Further application of the approach to practical problems through concrete examples is presented together with suggestions for numerical implementation. The numerical computations also demonstrate the stability of the numerical scheme with parameters chosen to satisfy the stability requirements.