We present a numerical study of the gas jet formed by the release of carbon dioxide at the sudden opening of a champagne bottle, typically sealed with a cork stopper, and its mixing with the exterior air. This investigation is stimulated by (and complements) the seminal experimental investigation carried out by Liger-Belair, Cordier & Georges (Science Advances, 5(9), 2019). Before the opening, the gas is at rest as confined by the stopper and the liquid, taken as quiescent throughout and having a constant level of a plane free surface. The whole flow configuration is assumed to be axisymmetric, and the contained gas and the air are taken as indistinguishable and thus perfectly mixable ideal gases. The resulting problem is governed by the Euler equations, governing the inviscid gas flow as suitably expressed in conservative form, the equation of motion of the stopper, and the kinematic and dynamic boundary conditions met on the bottle walls and the stopper surface. There they account for the propagation of the cork by the pressure forces exerted by the flow and, as long as the stopper has not completely left the bottleneck, the normal and Coulomb friction forces due to its sliding along the latter. The compressibility of the cork is considered through its typical hyperelastic constitutive behavior. We employed the open-source gas dynamics simulation package Clawpack to solve the so obtained fluid-structure interaction problem. Some modifications of the underlying Godunov's and Roe's schemes proved necessary so as to resolve the fast axial motion of the cork with due accuracy. As a definite highlight, in good agreement with the experimental findings, our results disclose the formation of Mach disk between the bottle opening and the freely moving cork as well as a precursor shock in front of it. We discuss the dependence of the solutions on the temperature of the contained gas and their potential impact on problems in the wider context of transitional ballistics.