Multiresolution adaptive space refinement in geophysical fluid dynamics simulation

Summary

We review part of the methodology for multiresolution adaptive solution of PDEs introduced by Alpert et al. (2002) in 1D (§2.1), and introduce a 2D generalization and implementation (§2.2). This methodology is similar to the spectral-element method (SEM, e.g., Fournier et al. 2004) in that it combines spectral accuracy with finite-element efficiency, but is not exactly SEM. We present 2D dynamical test cases (§2.3) that exhibit decreasing range of active scales (Heat Eq.), or else increasing range due to strong nonlinearities (Burgers Eq.).We conclude by showing that our methodology adapts to such evolving phenomena in these PDEs (§3), thereby saving computational cost, while preserving a high preselected representation accuracy per time step.