PIMS-SFU Applied & Computational Math Seminar: Brian Wetton

Adaptive time stepping methods for meta-stable dynamics of the Allen Cahn and Cahn Hilliard equations are investigated in the spatially continuous, semi-discrete setting. Predicted behaviour of the number of time steps with specified local error tolerance $\sigma$ based on the local truncation error and its dependence on the order parameter $\epsilon$  in the equations is verified in computational studies. A number of first and second order methods are considered. It is seen that in this setting, some methods, including but not limited to so-called energy stable methods, require asymptotically (as $\epsilon \rightarrow 0$) more time steps. It is shown that having the dominant local truncation error a pure time derivative is necessary for optimal number of time steps. Further, we introduce the concept of asymptotic consistency and show in this context that first order Backward (Implicit) Euler has a particularly desirable property. This last result is obtained with a formal asymptotic argument backed by computational evidence. This is joint work with Xinyu Cheng and Keith Promislow.