Domain decomposition (DD) methods provide a natural computational framework for multiscale multiphysics problems and a powerful tool for parallel numerical simulation of large-scale problems. As many physical and engineering processes are described by evolution partial differential equations, extensions of DD methods to dynamic systems (i.e., those changing with time) have been a subject of great interest. Moreover, for applications in which the time scales vary considerably across the whole domain due to changes in the physical properties or in the spatial grid sizes, it is critical and computationally efficient to design DD methods which allow the use of different time step sizes in different subdomains.
In this talk, we first introduce mathematical concepts of DD methods for evolution equations, then present DD-based time-stepping methods for the rotating shallow water equations discretized on spatial meshes with variable resolutions. Two different approaches will be considered: the first approach is a fully explicit local time-stepping algorithm based on the Strong Stability Preserving Runge-Kutta (SSP-RK) schemes, which allows different time step sizes in different regions of the computational domain. The second approach is the so-called Localized Exponential Time Differencing (LETD) method, which makes possible the use of much larger time step sizes compared to explicit schemes and avoids solving nonlinear systems as required in an implicit time discretization. Numerical results on various test cases will be presented to demonstrate the performance of the proposed methods.
BIO: Thi Thao Phuong Hoang is an assistant professor in the department of Mathematics and Statistics at Auburn University. She received her Ph.D. in Applied Mathematics in 2013 from the Université Pierre et Marie Currie (Paris VI), France. From 2014 to 2016, she worked at the Ho Chi Minh City University of Education, Vietnam as an assistant professor of Mathematics. Since 2017, she was a postdoctoral fellow in the Interdisciplinary Mathematics Institute, University of South Carolina Columbia before joining Auburn University in 2018.
Dr. Hoang's research areas include applied mathematics, numerical analysis, and scientific computing. Her research focuses on the development and analysis of efficient numerical methods for multiscale multiphysics problems arising in fluid mechanics, geophysics, and materials science. In particular, she has worked extensively on parallel numerical algorithms based on domain decomposition and local time stepping with applications related to flow and transport in fractured porous media, and coastal ocean modeling. Her research has been funded by the National Science Foundation (NSF) and Auburn University Intramural Grants Program. She is the recipient of a 2021 Faculty Early Career Development (CAREER) award from the NSF.