Full space-time adaptive method based on collocation strategy and implicit multirate time stepping

In this paper, we present a full space-time Adaptive Fup Collocation Method (AFCM) for solving initial-boundary value problems with particular application on advection-dominated advection-diffusion problems. The spatial adaptive strategy dynamically changes the computational grid at each global time step, while the adaptive time-marching algorithm uses different local time steps for different collocation points based on temporal accuracy criteria. Contrary to the existing space-time adaptive methods that are based on explicit temporal discretization and scale-dependent local time stepping, the presented method is implicit and resolves all spatial and temporal scales independently of each other. This means that smaller local time steps are used only for spatial zones where temporal solution changes are intensive, which are not necessarily the zones with finest spatial discretization. In this manner, the computationally efficient full space-time adaptive strategy accurately resolves small-scale features and controls numerical error and spurious oscillations. The efficiency and accuracy of AFCM are verified with some classical one- and two-dimensional benchmark test cases.