A WELL-BALANCED ASYMPTOTIC PRESERVING SCHEME FOR THE TWO-DIMENSIONAL SHALLOW WATER EQUATIONS OVER IRREGULAR BOTTOM TOPOGRAPHY

A new well-balanced asymptotic preserving scheme for two-dimensional low Froude number shallow water flows over irregular bottom is developed in this study. The bed-slope terms in the low Froude number regime are nontrivial since their stiffness has the same order as the gravity waves, they change the flow behavior in the low Froude number regime, and thus require special treatment when developing a numerical scheme to ensure such terms will not introduce high order numerical diffusion and spurious waves. To this end, the governing system is reformulated to obtain the well-balanced property. Since the system is stiff in the low Froude number flow regime, conventional explicit numerical schemes are extremely inefficient and often impractical. In order to overcome such difficulties, an asymptotic preserving scheme is developed by splitting the flux into a slow nonlinear part and fast linear part first, then approximating the slow dynamics explicitly using an explicit shock capturing scheme while estimating the fast dynamics implicitly. Using in space the linear piecewise reconstruction with minmod limiter for the shock explicit capturing scheme and central difference method for implicit derivatives, and in time the second order implicit-explicit Runge-Kutta methods, the second order accuracy of the proposed scheme is achieved. It is proved that the proposed numerical schemes are asymptotically consistent and stable uniformly with respect to small Froude number. Several numerical experiments are conducted to demonstrate the performance of the proposed asymptotic preserving numerical methods.