Transport modelling and in particular transport assignment is a wellknown bottleneck in computation cost and time for urban system models. The use of Transport Analysis Zones (TAZ) implies a trade-off between computation time and accuracy: practical computational constraints can lead to concessions to zone size with severe repercussions for the quality of the transport representation in urban models. This paper investigates how a recently developed geographical topology called adaptive zoning can be used to obtain more favorable trade-offs between computational cost and accuracy than traditional TAZ. Adaptive zoning was developed specifically for representing spatial interactions; it makes use of a nested zone hierarchy to adapt the model resolution as a function of both the origin and destination location. In this paper the adaptive zoning method is tied to an approach to trip assignment that uses high spatial accuracy (small zones) at one end of the route and low spatial accuracy (large zones) at the other end of the route. Opportunistic use of either the first or second half of such routes with asymmetric accuracy profiles leads to a method of transport assignment that is more accurate than traditional TAZ based assignment at reduced computational cost. The method is tested and demonstrated on the well-known Chicago Regional test problem. Compared with an assignment using traditional zoning, an adaptive-zoning-based assignment that uses the same computation time reduces the bias in travel time by a factor 16 and link level traffic volume RMSE by a factor 6.4.