Nicole Beisiegel, Cristóbal E. Castro, Jörn Behrens

Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for example for tsunami, hurricane or typhoon prediction. Theoretical insight for mesh-based numerical methods, however, is largely restricted to uniform meshes as they allow for a traditional definition of convergence and computational complexities linearly multiply with mesh resolution. To gain more insight into adaptive meshes and build a theoretical framework for their assessment, we present and discuss a number of mesh metrics that we apply to simulations on an adaptive triangular mesh. The latter is driven by physics-based refinement indicators that capture relevant physical processes and determine the areas of mesh refinement/coarsening. The mesh metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time and allow for a novel definition of convergence which can be used to demonstrate a convergence speed-up of adaptive simulations. Furthermore, we obtain numerical evidence for computational overhead as well as the distribution of the numerical errors across the various mesh resolutions.