A positivity preserving central scheme for shallow water flows in channels with wet-dry states

Abstract

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Open Water Flow in a Wet/Dry Multiply-Connected Channel Network: A Robust Numerical Modeling Algorithm

“…A sufficient condition for this is to ensure that at each time step no more water outflows from a cell than there is at the moment. For the positivity, our result is similar to the one obtained in (Balbás and Hernandez-Duenas 2014).…”

“…We expect that the scheme (51), (52), (54) has first-order accuracy in time and second-order accuracy in space. We consider the test proposed in (Balbás and Hernandez-Duenas 2014) to check second-order accuracy of the scheme in space for smooth flows by evaluation of L 1 error over the successively thinner grids.…”

“…When the perturbation propagates, part of it reflects back and part is transmitted, inundating the right side and leaving the area through the right boundary. According to (Balbás and Hernandez-Duenas 2014), the bottom topography is described by a piecewise spline. At the beginning, we Table 1 Accuracy test for the scheme in smooth regions.…”

“…Among them, central (Kurganov and Tadmor 2000) and centralupwind (Kurganov and Levy 2002) schemes are widely used in a wide range of applications due to their simplicity, efficiency, and robustness. For the shallow water flow in open channels, these schemes were applied in Balbás and Karni (2009), Balbás and Hernandez-Duenas (2014), Hernandez-Duenas and Beljadid (2016). Balbás and Karni (2009) presented a central finite-volume scheme of second-order accuracy to simulate one-dimensional shallow water flows along channels with non-uniform rectangular crosssections and bottom topography.…”

“…Balbás and Karni (2009) presented a central finite-volume scheme of second-order accuracy to simulate one-dimensional shallow water flows along channels with non-uniform rectangular crosssections and bottom topography. Balbás and Hernandez-Duenas (2014) extended this scheme for channels with cross-sections of arbitrary shape and bottom topography. A central-upwind scheme with artificial viscosity was proposed by Hernandez-Duenas and Beljadid (2016) for shallow water flows in channels with arbitrary geometry and variable topography.…”