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Moving-Water Equilibria Preserving Partial Relaxation Scheme for the Saint-Venant System

Xin Liu Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055,China, and Numerical Environmental Prediction Section, Canadian Meteorological Centre, Environ-ment and Climate Change Canada, Dorval, QC, H9P 1J3, Canada Xin Chen Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China, and De-partment of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China Shi Jin School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao TongUniversity, Shanghai, 200240, China Alexander Kurganov Department of Mathematics and SUSTech International Center for Mathematics, Southern Uni-versity of Science and Technology, Shenzhen, 518055, China Tong Wu Mathematics Department, Tulane University, New Orleans, LA 70118 Hui Yu Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China

Numerical Analysis and Scientific Computing mathscidoc:2205.25018

SIAM Journal on Scientific Computing, 42, (4), A2206-A2229, 2020.7
We develop a new moving-water equilibria preserving numerical scheme for the Saint-Venant system. The new scheme is designed in two major steps. First, the geometric source term is incorporated into the discharge flux, which results in a hyperbolic system with a global flux. Second, the discharge equation is relaxed so that the nonlinearity is moved into the stiff right-hand side of the added auxiliary equation. The main advantages of the new scheme are that (i) no special treatment of the geometric source term is required, and (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables, as it must be done in the existing alternative methods. We also develop a hybrid numerical flux, which helps to handle various flow regimes in a stable manner. Several numerical experiments are performed to verify that the proposed scheme is capable of exactly preserving general moving-water steady states and accurately capturing their small perturbations.
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@inproceedings{xin2020moving-water,
  title={Moving-Water Equilibria Preserving Partial Relaxation Scheme for the Saint-Venant System},
  author={Xin Liu, Xin Chen, Shi Jin, Alexander Kurganov, Tong Wu, and Hui Yu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520143101309967305},
  booktitle={SIAM Journal on Scientific Computing},
  volume={42},
  number={4},
  pages={A2206-A2229},
  year={2020},
}
Xin Liu, Xin Chen, Shi Jin, Alexander Kurganov, Tong Wu, and Hui Yu. Moving-Water Equilibria Preserving Partial Relaxation Scheme for the Saint-Venant System. 2020. Vol. 42. In SIAM Journal on Scientific Computing. pp.A2206-A2229. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220520143101309967305.
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