We consider the isothermal Euler equations without friction that simulate gas flow through a pipe. We consider the problem of boundary stabilisation of this system locally around a given stationary state. We present a feedback law that is linear in the physical variables and yields exponential decay of the system state. For the numerical solution of hyperbolic systems of conservation laws, the Jin-Xin relaxation scheme can be used. Therefore, we also consider the boundary stabilisation of the relaxation system by the linear Riemann feedback and present numerical examples that show the rapid exponential decay of the stabilised system.