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A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional saint-venant system

Communications in Computational Physics 2023年第5期33卷 1432-1465页

作者： Xi Chen Alexander Kurganov Department ofMathematics Harbin Institute of TechnologyHarbin150001China and Department of MathematicsSouthern University of Science and TechnologyShenzhen518055China Department of Mathematics SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material DesignSouthern University of Science and TechnologyShenzhen518055China

We develop a new moving-water equilibria preserving partial relaxation(PR)scheme for the two-dimensional(2-D)saint-venant systemof shallowwater *** new scheme is a 2-D generalization of the one-dimensional(1-D)PR scheme recently proposed in[***,***,***,***,***,***.,42(2020),pp.A2206–A2229].Our scheme is based on the PR approximation,which is designed in two ***,the geometric source terms are incorporated into the discharge fluxes,which results in a hyperbolic system with global ***,the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary *** obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finitevolume method combined with an efficient semi-implicit ODE *** 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 *** 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.

关键词： saint-venant system of shallow water equations partial relaxation scheme wellbalanced method steady-state solutions(equilibria) moving-water and still-water equilibria

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