This article discusses a numerical method for solving the direct and inverse problems of the gas-lift process of oil production described by one-dimensional Navier-Stokes equations for compressible gas. To solve the direct problem, a family of difference schemes is constructed and the correctness of this difference problem depending on the parameter is investigated. The solution to the inverse problem is reduced to an optimal control problem. The objective functional is constructed using an additional condition. To minimize the objective functional, a gradient method is used. The gradient of the objective functional is determined through the solution of the conjugate problem, which contains valuable information about the solution of the direct problem. And the conjugate problem itself is constructed using the Lagrange identity and the condition of equality to zero of the integral terms. The conjugate problem is retrospective, since the additional condition on the volumetric gas flow rate and pressure are set at a certain point in time. Iterations determine the initial conditions for the volumetric gas flow rate and pressure through the solution of the conjugate, retrospective problem. A computational experiment was conducted using the developed algorithm, which confirms that the initial values ​​of the volumetric gas flow rate and pressure can be determined with the required accuracy using the specified additional condition. The developed algorithm can be used to plot a curve for the gas-lift process performance.