In this paper, we consider the Euler equations with spherical symmetry inoutside a region of the center. By assuming an outing absorbing boundary and within the leading term with respect to the amplitude of wave strength, we prove that there exists a uniform bound for the approximate solutions constructed by Glimm's scheme. This implies that the main difficulty left for the full nonlinear system whenx≥1 is to estimate the infinite reflections of waves inside the region, whereis the supremum of the absolute value of the characteristics under consideration.