A new efficient implicit scheme for discretising the stiff friction terms in the shallow water equations

Discretisation of the friction terms to ensure numerical stability and accuracy remains to be challenging for the development of robust numerical schemes to solve the shallow water equations (SWEs), particularly for applications involving very shallow flows (e.g. overland flows and wet/dry fronts) over complex domain topography. The key challenge is to ensure relaxation of the flow towards an equilibrium state characterised by the balance between friction and gravity in a computationally efficient way. To overcome this numerical challenge, this paper proposes a novel approach for discretising the friction source terms in the SWEs in the context of an explicit finite volume method. The overall numerical scheme adopts the HLLC Riemann solver and surface reconstruction method (SRM) to explicitly discretise the flux and bed slope source terms. Whilst a fully implicit scheme is used to handle the friction source terms, solution to the implicit formulation is analytically derived to explicitly update the flow variables. Compared with the existing approaches, the proposed scheme effectively resolves the issue associated with stiff relaxation without necessity to use an iteration method and it supports efficient simulation using time steps controlled only by the Courant-Friedrichs-Levy (CFL) condition. The current friction term discretisation scheme is not coupled with flux and bed slope calculation and therefore may be readily implemented in any other explicit finite volume SWE models. After being successfully validated against two benchmark tests with analytical solutions, the resulting new SWE model is applied to reproduce a rainfall-flooding event in the Upper Lee catchment in the UK.