An Asymptotic Preserving Scheme for Low Froude Number Shallow Flows

We present an asymptotic preserving (AP), large time-step scheme for the shallow water equations in the low Froude number limit. Based on a multiscale asymptotic expansion, the momentum fluxes are split into a nonstiff and a stiff part. A semi-implicit discretisation, where the nonstiff terms are treated explicitly and stiff terms implicitly in time, is crucial to achieve the AP property. A combination of the semi-discrete mass and momentum equations leads to an elliptic equation for the water height at the new time-level. With the aid of this, the momentum can be update explicitly using a large timestep which solely determined by the nonstiff characteristic speeds. The second order accuracy of the scheme is based on Runge-Kutta and Crank-Nicolson time-stepping procedures and MUSCL-type reconstructions. The numerical results clearly demonstrate the accuracy and robustness of the scheme and its efficacy to compute very low Froude number flows. 2010 Mathematics Suject Classification Primary 35L65, 76B15, 76M45; Secondary 65M08, 65M06 Shallow water equations, low Froude number limit, stiffness, semi-implicit time discretisation, flux decomposition, asymptotic preserving schemes