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Volume 33, Issue 5
A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System

Xi Chen & Alexander Kurganov

Commun. Comput. Phys., 33 (2023), pp. 1432-1465.

Published online: 2023-06

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  • Abstract

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.

  • AMS Subject Headings

76M12, 65M08, 86-08, 86A05, 35L65

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COPYRIGHT: © Global Science Press

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@Article{CiCP-33-1432, author = {Chen , Xi and Kurganov , Alexander}, title = {A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {5}, pages = {1432--1465}, abstract = {

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} }
TY - JOUR T1 - A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System AU - Chen , Xi AU - Kurganov , Alexander JO - Communications in Computational Physics VL - 5 SP - 1432 EP - 1465 PY - 2023 DA - 2023/06 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0319 UR - https://global-sci.org/intro/article_detail/cicp/21767.html KW - Saint-Venant system of shallow water equations, partial relaxation scheme, well-balanced method, steady-state solutions (equilibria), moving-water and still-water equilibria. AB -

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.

Chen , Xi and Kurganov , Alexander. (2023). A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System. Communications in Computational Physics. 33 (5). 1432-1465. doi:10.4208/cicp.OA-2022-0319
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