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 - <p style="text-align: justify;">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.</p>