This paper defines and studies a simple, efficient method for discretizing pipeline equations in time.
In many ways the solutions of pipe flow equations look like the solutions of nonlinear diffusion equations, so it is natural that schemes that work well for diffusion problems might have analogs for pipe flow problems.
The method presented here is based on backward Euler (BE). To advance the solution from time t to time t+dt it uses a single step of BE with step size dt and two steps of BE with step size ????/2. The results are then combined to give a second-order correct scheme. This process is called step doubling with local extrapolation (SDoLE) and has been rigorously analyzed for nonlinear diffusion problems in the context of Galerkin spatial discretizations.  Here a collocation scheme is the underlying discretization in space and time.
The backward Euler that forms the basis of this method is done using a linearization technique instead of solving a nonlinear system. This is extended to minimize the computational costs associated with evaluation of the nonlinearities in the equations. The resulting discretization technique is called affine step doubling with local extrapolation (ASDoLE).
The properties of the method are explained by looking first at a scalar ODE and then by presenting examples based on pipeline operation. A single pipe case is given followed by a simple gun barrel example.