Three-Layer Approximation of Two-Layer Shallow Water Equations

Abstract

Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.

Keywords:

• two-layer shallow water equations
• central-upwind scheme
• well-balanced scheme
• conditional hyperbolicity

AMS Subject Classification:

• 76E20
• 76M12
• 65M08
• 86-08
• 35L65
• 35L67