Abstract <p>We study the one-dimensional problem of an infinite tube which is open in the right and there is a piston put at one end in the left. Due to the fact that the computational domain is finite while the domain of the problem is infinite, the numerical result is affected by the presence of the reflected wave appears when the shock travels to the right and interacts with the right boundary. Thus there is the need of implementing a non-reflecting boundary condition in order to eliminate as much as possible the affection of reflected wave. In this paper, we use Euler equations in the mass Lagrangian coordinates as the govern equations and apply finite volume method to compute the numerical solution. In order to deal with the reflected wave, we use the Burgers-like equation in an additional computational domain. The numerical results we obtain show that the numerical error is reduced significantly.</p>