► Numerical simulation of the start-up flow of thixotropic drilling fluids. ► The thixotropy model accounts for both viscous-plasticity and material elasticity. ► Thixotropy properties, compressibility and Reynolds number affect pressure peaks. This study presents a mathematical model to simulate the start-up flow of drilling fluids in drill pipes. The compressible transient flow approach is based on one-dimensional mass and momentum conservation equations, an equation of state and a thixotropic fluid model. In contrast with prior studies that view gel breaking as a visco-plastic phenomenon, thixotropy is approached as an elasto-visco-plastic problem. Such thixotropy model is fit to rheometer data with reasonable agreement. The start-up flow is then simulated, and the results are compared to Newtonian, Bingham and visco-plastic fluid flow models. It can be anticipated that the pressure overshoots observed in start-up flows depend not only on the thixotropy properties of the fluid but also on the flow compressibility and Reynolds number. In addition, the main differences between the visco-plastic and elasto-visco-plastic thixotropy approaches take place at low shear rates; the higher the shear rates, the more similar are the results. Finally, as expected, the pressure overshoots of the thixotropic fluids are not always larger than those of the Bingham and Newtonian counterparts.