I will describe a method for approximating density operators of 1D systems called "DMT". When combined with a standard framework for time evolution (TEBD), DMT makes possible simulation of the dynamics of large thermalizing systems to long times (with computation time linear in system size and time). As a benchmark, I will apply DMT to the dynamics of the ETH random-field Heisenberg model. Time permitting, I will also discuss its application to the prethermal regime of a spin system with a high-frequency drive, where we see not only the expected slow Floquet heating but also clear diffusive hydrodynamics.