Retaining desirable properties in discretising a travel-time model

A recent paper introduced a new whole-link travel time model and showed that it has various desirable properties, including a first-in-first-out (FIFO) property, causality and consistency with the usual static model when flows are constant. The model is formulated as a continuous-time first-order differential equation, which does not have a general analytical solution but can be solved (approximately) numerically by forward or backward discrete-time differencing methods. Here we show that if the step sizes are not arbitrarily small then the solutions obtained by the usual differencing methods do not always preserve FIFO. In view of that, we introduce a new differencing method and prove that it always preserves FIFO and the other desirable properties exhibited by the continuous-time model. In numerical examples we illustrate how the new discrete-time differencing model eliminates FIFO violations, illustrate convergence of a solution process for the new model, and illustrate how various inflow patterns affect FIFO under the old and new differencing methods.