Essentially Non-Oscillatory Schemes over Unstructured Meshes Based on Weighted Least Squares

Xiangmin (Jim) Jiao
Seminar

CT

Argonne National Laboratory
Building 240/Room 1406
Host Vijay Mahadevan
Event Sponsor Mathematics and Computing Science Seminar

Weighted least squares (WLS) is a unified framework for accurate and stable numerical discretizations over unstructured meshes and point clouds. In our recent works, we have shown that WLS can deliver the same order of accuracy and better stability than interpolation-based techniques, and we have successfully applied WLS to develop accurate numerical methods, including high-order reconstructions and integrations over discrete surfaces, and generalized finite difference methods over unstructured meshes. In this talk, we describe our recent work in applying the WLS framework to construct a systematic extension of the WENO (Weighted Essentially Non-Oscillatory) schemes to unstructured meshes. Our approach, called LS-WENO, can deliver higher accuracy and better stability than the traditional WENO schemes over structured meshes. More importantly, LS-WENO generalizes to unstructured meshes to deliver accurate and stable schemes, while the previous generalizations were shown to be unstable. We will also present an extension of the finite element methods within the WLS framework, which enables the integration of LS-WENO into the finite element framework, especially over adaptive unstructured meshes. We will also discuss some remaining challenges and future research directions.

Short Bio:

Dr. Xiangmin Jiao is an associate professor in the Department of Applied Mathematics and Statistics and an adjunct associate professor in the Department of Computer Science of Stony Brook University. He received his Ph.D. in computer science in 2001 from University of Illinois at Urbana-Champaign (UIUC). After receiving his Ph.D., he was a research scientist at the Center for Simulation of Advanced Rockets (CSAR) at UIUC and then a visiting assistant professor at Georgia Institute of Technology, before joining Stony Brook in 2007. His current research interests focus on developing efficient and robust algorithms and implementations for numerical discretizations over complex geometries, dynamic surfaces, optimal multigrid solvers, applied computational and differential geometry, multiphysics coupling, and their applications in various engineering and physical sciences.