Nonstaggered central scheme with steady-state discretization for solving the open channel flows via the flux globalization

Abstract

The paper proposed a second-order steady-state-preserving nonstaggered central scheme for solving one-layer and two-layer open channel flows via the flux globalization. The global flux transforms the model into the homogeneous form, avoiding the complex discretization of the source terms. However, when the traditional appropriate quadrature rule discrete the global variables, the scheme tends to maintain only the moving-water equilibrium but not the “lake at rest” equilibrium. This paper proposes a new discretization method, the steady-state discretization (SSD) method of global variables, so that not only the still-water equilibrium can be maintained, but also the moving-water equilibrium, i.e., the discharge, the energy and the global flux are equilibrium. The scheme also ensures that the cross-sectional wet area is positive by introducing a “draining” time-step technique. Numerical experiments verify that the scheme is well-balanced, positivity-preserving and robust when flowing through open channel flows under the continuous or discontinuous bottom topography and channel width, and exactly capturing small perturbations and propagating interfaces of the steady-state solution.