An efficient and stable hydrodynamic model with novel source term discretization schemes for overland flow and flood simulations

Abstract

Numerical models solving the full 2‐D shallow water equations (SWEs) have been increasingly used to simulate overland flows and better understand the transient flow dynamics of flash floods in a catchment. However, there still exist key challenges that have not yet been resolved for the development of fully dynamic overland flow models, related to (1) the difficulty of maintaining numerical stability and accuracy in the limit of disappearing water depth and (2) inaccurate estimation of velocities and discharges on slopes as a result of strong nonlinearity of friction terms. This paper aims to tackle these key research challenges and present a new numerical scheme for accurately and efficiently modeling large‐scale transient overland flows over complex terrains. The proposed scheme features a novel surface reconstruction method (SRM) to correctly compute slope source terms and maintain numerical stability at small water depth, and a new implicit discretization method to handle the highly nonlinear friction terms. The resulting shallow water overland flow model is first validated against analytical and experimental test cases and then applied to simulate a hypothetic rainfall event in the 42 km2 Haltwhistle Burn, UK.

Numerical stability analysis of the local inertial equation with semi- and fully implicit friction term treatments: assessment of the maximum allowable time step

“…The behavior of the momentum equation for large time step (∆t → +∞) is a key for assessing stability of numerical schemes for the equations of shallow water flows since it effectively characterizes qualitative properties of numerical solutions for not small ∆t and small water depths [19]. For the large time step limit (∆t → +∞) that represents a practical situation where the temporal resolution is not sufficiently fine, q k+1/2 i+1/2 with the fully implicit treatment approaches the equation…”

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

“…The required Riemann states are obtained using the surface reconstruction method (SRM) as proposed by Xia et al (2017) . SRM firstly reconstructs the water surface elevations at the left and right-hand sides of a given cell interface.…”

“…As pointed out in Xia et al (2017) , inappropriate discretisation of the friction source terms may result in 1) numerical instability that leads to the use of prohibitive small time steps, and 2) inaccurate prediction of flow velocities.…”

“…Liang et al, 2007;Cea et al, 2010;Song et al, 2011;Costabile et al, 2013;Busaman et al, 2015;Cea and Blade, 2015;Rousseau et al, 2015;Singh et al, 2015 ) express the friction terms as the product of the velocity in the current time step and that in the new time step to obtain an explicit formula. Although these schemes may effectively avoid the numerical instability caused by the stiff friction terms, they commonly relax the flows to a wrong steady state, which may consequently lead to incorrect simulation results ( Xia et al, 2017 ).…”

“…the kinematic wave number, is a case-dependent parameter, which restricts the robustness of the method for wider applications. More recently, Xia et al (2017) introduced a fully implicit friction discretisation scheme that can effectively relax the flow towards the correct steady state and allow the use of CFL time step even when it is much bigger than the relaxation time scale. Different from the aforementioned schemes that reformulate the equations into an explicit form, Xia et al (2017) directly solved the implicit friction equations using a Newton-Raphson iteration method.…”