Electrons can be accelerated by their interaction with nonlinearly saturated electrostatic waves up to speeds with which they can undergo diffusive acceleration across supernova remnant shocks. Here, we model this wave-electron interaction by particle-in-cell and Vlasov simulations. We find that the lifetime of the saturated wave is considerably longer in the Vlasov simulation, due to differences in how these simulation methods approximate the plasma. Electron surfing acceleration which requires a stable saturated wave may thus be more important for electron acceleration at shocks than previously thought. For beam speeds above a critical value, which we estimate here, both simulation codes exclude surfing acceleration due to a rapid wave collapse.