Non-reflecting boundary condition for the problem with a semi-infinite tube

We study a 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 a reflected wave that appears when a shock travels to the right and interacts with the right boundary. Thus there is a need of implementing a non-reflecting boundary condition in order to eliminate as much as possible the effect of the reflected wave. In this paper, we use the Euler equations in mass Lagrangian coordinates as the governing equations and apply a finite volume method to compute the numerical solution. In order to deal with the reflected wave, we use a Burgers-like equation in an additional computational domain. The numerical results we obtain show that the numerical error is reduced significantly.