One-dimensional models based on mass and momentum conservation equations are often used to describe the dynamics of gas flows in gas pipeline systems. These equations, supplemented with empirical and semi-empirical relations, account for various physical processes such as friction and heat transfer. Nodal models are used to model the interactions between multiple pipes, which rely on the principles of conservation of mass and energy. The functional components of the gas transmission network are modeled by functional dependencies. A computational algorithm based on linearization of a purely implicit scheme and finite element approximation over space is proposed. The matrix rows corresponding to the new variables are used to specify the equations realizing the functional components of the gas transportation system and the conditions at the pipe joints. The capabilities of the developed algorithm are demonstrated on different test problems. Special attention is paid to the convergence of the numerical solution when the time step decreases.