- Journal Home
- Volume 36 - 2024
- Volume 35 - 2024
- Volume 34 - 2023
- Volume 33 - 2023
- Volume 32 - 2022
- Volume 31 - 2022
- Volume 30 - 2021
- Volume 29 - 2021
- Volume 28 - 2020
- Volume 27 - 2020
- Volume 26 - 2019
- Volume 25 - 2019
- Volume 24 - 2018
- Volume 23 - 2018
- Volume 22 - 2017
- Volume 21 - 2017
- Volume 20 - 2016
- Volume 19 - 2016
- Volume 18 - 2015
- Volume 17 - 2015
- Volume 16 - 2014
- Volume 15 - 2014
- Volume 14 - 2013
- Volume 13 - 2013
- Volume 12 - 2012
- Volume 11 - 2012
- Volume 10 - 2011
- Volume 9 - 2011
- Volume 8 - 2010
- Volume 7 - 2010
- Volume 6 - 2009
- Volume 5 - 2009
- Volume 4 - 2008
- Volume 3 - 2008
- Volume 2 - 2007
- Volume 1 - 2006
Commun. Comput. Phys., 33 (2023), pp. 1432-1465.
Published online: 2023-06
Cited by
- BibTex
- RIS
- TXT
We develop a new moving-water equilibria preserving partial relaxation (PR) scheme for the two-dimensional (2-D) Saint-Venant system of shallow water equations. The new scheme is a 2-D generalization of the one-dimensional (1-D) PR scheme recently proposed in [X. Liu, X. Chen, S. Jin, A. Kurganov, and H. Yu, SIAM J. Sci. Comput., 42 (2020), pp. A2206–A2229]. Our scheme is based on the PR approximation, which is designed in two steps. First, the geometric source terms are incorporated into the discharge fluxes, which results in a hyperbolic system with global fluxes. Second, the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary equation. The obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finite-volume method combined with an efficient semi-implicit ODE solver. The new 2-D PR scheme inherits the main advantages of the 1-D PR scheme: (i) no special treatment of the geometric source terms is required, (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables. The performance of the proposed PR scheme is illustrated on a number of numerical examples, in which we demonstrate that the PR scheme not only capable of exactly preserving quasi 1-D moving-water steady states and accurately capturing their small perturbations, but can also handle genuinely 2-D steady states and their small perturbations in a non-oscillatory manner.
}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0319}, url = {http://global-sci.org/intro/article_detail/cicp/21767.html} }We develop a new moving-water equilibria preserving partial relaxation (PR) scheme for the two-dimensional (2-D) Saint-Venant system of shallow water equations. The new scheme is a 2-D generalization of the one-dimensional (1-D) PR scheme recently proposed in [X. Liu, X. Chen, S. Jin, A. Kurganov, and H. Yu, SIAM J. Sci. Comput., 42 (2020), pp. A2206–A2229]. Our scheme is based on the PR approximation, which is designed in two steps. First, the geometric source terms are incorporated into the discharge fluxes, which results in a hyperbolic system with global fluxes. Second, the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary equation. The obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finite-volume method combined with an efficient semi-implicit ODE solver. The new 2-D PR scheme inherits the main advantages of the 1-D PR scheme: (i) no special treatment of the geometric source terms is required, (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables. The performance of the proposed PR scheme is illustrated on a number of numerical examples, in which we demonstrate that the PR scheme not only capable of exactly preserving quasi 1-D moving-water steady states and accurately capturing their small perturbations, but can also handle genuinely 2-D steady states and their small perturbations in a non-oscillatory manner.