Many good time stepping methods have been developed over the decades that are well-suited to all sorts of ODEs and DAEs. If one were to implement multiphysics solvers from scratch, one would probably want to consider the use of methods that are well-suited for multirate and DAE problems. In practice, however, code complexity issues and the difference between solving ODEs and spatially-discretized PDEs impose constraints we must consider. In this talk, I will outline my experience with incorporating modern time steppers into PDE solvers. Specifically, I will consider as an example the ASPECT code (short for ""Advanced Solver for Planetary Evolution, Convection, and Tectonics"") that solves problems relating to the motion of material, energy, and composition in both the Earth crust and the mantle. This is described by a coupled system that involves instantaneous Stokes flow, the advection and diffusion of a temperature field, and the advection and reaction of chemical compositions or other quantities. ASPECT can also simulateously solve for the deformation of the Earth surface, as well as advect particles with the flow field. The resulting time stepping scheme is diverse, but for the most part only uses home-grown time steppers that, for historical reasons, are not well integrated. I will discuss what we do and why, as well as what we do not do and why not. I will also discuss what I think better solutions could be.