Mathematical modeling and numerical simulations of multiscale multiphysics processes are of great importance, yet highly challenging as various processes occur at different scales and are coupled together. The issues are more crucial when dealing with complex large-scale systems (for example, those arising in ocean and coastal modeling). The goal of this project is to advance the efficiency and fidelity of local time-stepping algorithms for multiscale multiphysics systems with application to multi-resolution simulations of large-scale geophysical flows. The developed algorithms will efficiently capture the wide range of scales in both space and time to produce accurate and robust simulations of these systems over a long period of time. The research plan is closely integrated with the educational activities of the project which include (i) developing curricular modules in computational mathematics at the Auburn University Summer Science Institute, an educational enrichment program for high school students, to provide young students early exposure to applied mathematics and inspire them to pursue a career in Science, Technology, Engineering and Mathematics (STEM); and (ii) providing interdisciplinary applied mathematics education and research training for both undergraduate and graduate students, including women and underrepresented minorities.
Technically, the Principal Investigator will develop accurate and effective hybrid local time-stepping algorithms based on nonoverlapping domain decomposition: on the one hand, explicit schemes with local time steps are used to model processes at small time scales without suffering a severe restriction on the time step size dictated by the global CFL condition. On the other hand, localized exponential time integrators are employed to enable large time step sizes for processes occurring at slow speeds, and to accelerate the computation of matrix exponentials and their products by performing these calculations locally and in parallel. Three main research objectives will be pursued: (i) development and analysis of nonoverlapping localized exponential time differencing methods for stiff nonlinear equations; (ii) study of hybrid local time-stepping algorithms for various heterogeneous problems; and (iii) application of these algorithms to the three-dimensional primitive equations for modeling ocean/atmosphere circulations.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
