Numerical Methods for the Nonlinear Shallow Water Equations线上短课程招生简章

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:2022-04-22 12:21


Yulong Xing (The Ohio State University)

Dr. Yulong Xing is a professor in the Department of Mathematics at the Ohio State University (OSU). He received his bachelor degree from University of Science and Technology of China in 2002, and Ph.D. in Mathematics from Brown University in 2006 under the supervision of Prof. Chi-Wang Shu. Prior to joining OSU, he worked as a Postdoctoral Researcher at Courant Institute, New York University, a staff scientist at Oak Ridge National Laboratory, a joint assistant professor at University of Tennessee Knoxville, and an assistant professor at University of California Riverside. He works in the area of numerical analysis and scientific computing, wave propagation, computational fluid dynamics. His research focuses on the design, analysis and applications of accurate and efficient numerical methods for partial differential equations arising from science and engineering problems. He has received a CAREER Award from the National Science Foundation.

助教:Zheng Sun (University of Alabama) 


Free surface flows often appear in ocean, engineering and atmospheric modelling. In many applications involving unsteady water flows where the horizontal length scale is much greater than the vertical length scale, the nonlinear shallow water equations (SWEs) are commonly used to model these flows. Research on effective and accurate numerical methods for their solutions has attracted great attention in the past two decades. In this series of lectures, we will talk about various numerical methods to solve the nonlinear shallow water equations. Some shallow water-related models and their numerical approximation will also be presented.

Lecture 1: We will start by presenting some applications of SWEs and the derivation of the SWEs under reasonable assumptions. For the homogeneous SWEs with no source term, how to analytically solve the Riemann problem will be discussed. Some first and second order numerical methods for the homogeneous SWEs will also be introduced.

Lecture 2: We will discuss the high order numerical methods for the homogeneous SWEs. High order finite difference and finite volume WENO methods will be discussed, as well as the finite element discontinuous Galerkin (DG) methods with suitable slope limiter procedure. Strong-stability-preserving Runge-Kutta methods will be presented as temporal discretization.

Lecture 3: We consider the SWEs with a source term due to the non-flat bottom topography. Well-balanced methods to exactly preserve the still-water or moving-water steady states will be discussed.

Lecture 4: Other high order structure-preserving methods for the SWEs, which preserve certain properties of the model in the discrete level, will also be discussed. For the SWEs with bottom topography, solving for the exact solution of Riemann problem is a challenging task, and we will review some existing work.

Lecture 5: Various temporal discretizations will be discussed for the SWEs to provide structure-preserving property. Some shallow water-related models, including the shallow water flows through channels, Ripa model and SWEs on the sphere, and their numerical approximation may be discussed.














 Numerical Methods for the Nonlinear Shallow Water Equations线上短课程招生简章