Techniques for Locally Adaptive Time Stepping Developed over the Last Two Decades

Adaptive mesh refinement techniques are well established and widely used for space discretizations. In contrast, local time stepping is much less used, and the corresponding techniques are less mature, needing delicate synchronization steps, which involve interpolation, extrapolation or projection. These operations can have adverse effects on the stability, and can also destroy important geometric properties of the scheme, like for example the conservation of invariants. We give here a survey on the intensive research performed in this direction over the last two decades. We explain multirate time stepping methods and multirate extrapolation methods stemming from the community of ordinary differential equations (ODEs), and also interpolation, extrapolation, projection and iteration techniques from the community of partial differential equations (PDEs). We select from each technique one influential paper, explain the technique in detail for a simple example, and summarize theoretical results obtained.