A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System (Communications in Computational Physics, 2023)
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 schemerecently 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 incorporatedinto 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 stiffright-hand side of the four added auxiliary equation. The obtained PR system is thennumerically integrated using a semi-discrete hybrid upwind/central-upwind finite-volume method combined with an efficient semi-implicit ODE solver. The new 2-D PRscheme inherits the main advantages of the 1-D PR scheme: (i) no special treatment ofthe geometric source terms is required, (ii) no nonlinear (cubic) equations should besolved 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 numberof numerical examples, in which we demonstrate that the PR scheme not only capableof exactly preserving quasi 1-D moving-water steady states and accurately capturingtheir small perturbations, but can also handle genuinely 2-D steady states and theirsmall perturbations in a non-oscillatory manner.
Keywords: Saint-Venant system of shallow water equations; partial relaxation scheme; well-balanced method; steady-state solutions (equilibria); moving-water and still-water equilibria.