A positivity preserving central scheme for shallow water flows in channels with wet-dry states∗
1 Department of Mathematics. California State University, Northridge. 18111 Nordhoff St. Northridge, CA 91330-8313, USA.
2 Department of Mathematics, University of Wisconsin – Madison. 480 Lincoln Dr., Madison, Wi 53706-1325, USA.
Received: 7 June 2012
We present a 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.
Mathematics Subject Classification: 76M12 / 35L65
Key words: Hyperbolic systems of conservation and balance laws / semi-discrete schemes / Saint-Venant system of Shallow Water equations / non-oscillatory reconstructions / channels with irregular geometry
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