Mohammad Farazmand

The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of relative quiescence. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. The proposed program will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.