Simulating an advancing fire front may be achieved within a Lagrangian or Eulerian framework. In the former, independently moving markers are connected to form a fire front, whereas in the latter, values representing the moving front are calculated at points within a fixed grid. Despite a mathematical equivalence between the two methods, it is not clear that both will produce the same results when implemented numerically. Here, we describe simulations of fire spread created using a level set Eulerian approach (as implemented in the wildland–urban interface fire dynamics simulator, WFDS) and a marker method (as implemented in FARSITE). Simulations of surface fire spread, in two different fuels and over domains of increasing topographical complexity, are compared to evaluate the difference in outcomes between the two models. The differences between the results of the two models are minor, especially compared with the uncertainties inherent in the modelling of fire spread.