A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System

We develop a new moving-water equilibria preserving partial relaxation(PR)scheme for the two-dimensional(2-D)Saint-Venant systemof shallowwater 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,andH.Yu,SIAMJ.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 finitevolume 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 equilibriumvariables.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.