Speaker: Dr. Ziyao Xu,Robert and Sara Lumpkins Postdoctoral Research Associate, University of Notre Dame
Date: Friday, November 15, 2024, 4:15 – 5:15 PM
Location: Science Engineering Hall (SCEN 404) in-person.
If you wish to receive the Zoom link, please contact Dr. Chen Liu
Title: A well-balanced conservative high-order alternative finite difference WENO (A-WENO) method for hyperbolic balance laws
Abstract:
In this work, we develop a well-balanced, conservative, high-order finite difference WENO method for hyperbolic balance laws. Our approach exactly preserves the moving-water equilibria of the shallow water equations with non-flat bottom topography and, more generally, any steady state of hyperbolic equations characterized by constant equilibrium variables. The proposed method consists of two key components. First, we reformulate the source terms in the balance laws into a flux-gradient form and discretize them using the same numerical flux as the true flux gradient to achieve the well-balanced property. Second, we reconstruct the equilibrium variables, which remain constant at steady state. To achieve high-order accuracy and avoid truncation errors when obtaining equilibrium variables, we build our scheme within the alternative finite difference WENO (A-WENO) framework, which operates on point values rather than cell averages. Special attention is given to ensure that the conservation property is not compromised when designing well-balanced discretizations for the source terms. We carefully analyze potential causes of non- conservative schemes in the discretization and explain why the discretized source term in our method is both conservative and simple. Extensive numerical tests are presented to validate the performance of the proposed method.
Biosketch:
Dr. Ziyao Xu is a Robert and Sara Lumpkins Postdoctoral Fellow in Applied and Com- putational Mathematics at the Department of Applied and Computational Mathematics and Statis- tics, University of Notre Dame. His research specializes in finite element methods, ENO/WENO methods, computational fluid dynamics, and fluid flow in porous media. Dr. Xu received his Ph.D. in Applied Mathematics at Brown University. Prior to that, he obtained his Master’s degree in Mathematics at Michigan Technological University and his Bachelor’s degree in Petroleum Engi- neering at China University of Petroleum (East China).
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