This paper proposes, for a fixed demand traffic network problem, a route travel choice adjustment process formulated as a projected dynamical system, whose stationary points correspond to the traffic equilibria. Stability analysis is then conducted in order to investigate conditions under which the route travel choice adjustment process approaches equilibria. We also propose a discrete time algorithm, the Euler method, for the computation of the traffic equilibrium and provide convergence results. The notable feature of the algorithm is that it decomposes the traffic problem into network subproblems of special structure, each of which can then be solved simultaneously and in closed form using exact equilibration. Finally, we illustrate the computational performance of the Euler method through various numerical examples.