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

Abstract

Abstract. The local inertial equation (LIE) as a simple mathematical model has been widely used for flood simulation. So far, the maximum allowable time step of the discretized LIE with the conventional semi-implicit scheme has been believed to follow the standard Courant-Friedrichs-Lewy condition. However, we demonstrate that this is not true from the viewpoint of a numerical stability analysis considering the model non-linearity. In addition, a fully-implicit variant of the scheme with higher stability is presented, indicating its practical advantages.

A consistent finite difference local inertial model for shallow water simulation

“…Stability analysis against a perturbed uniform flow is carried out to quantify stability conditions of the three schemes. The flow considered here is uni-directional, and the 1-D counterpart of the LIM (q = 0 and ∂h ∂ y = ∂z ∂ y = 0) is employed (Tanaka and Yoshioka, 2017). The initial condition is a uniform flow on a slope having a constant gradient.…”

“…It has been stated that the friction slope terms in 2and 3should be handled in a semi-implicit manner for numerical stability (Bates et al, 2010;de Almeida and Bates, 2013). However, recently, Tanaka and Yoshioka (2017) found that this discretization becomes unstable even when the classical Courant-Friedrichs-Lewy (CFL) condition is satisfied. An implicit scheme was developed as an alternative to the semi-implicit one (Tanaka and Yoshioka, 2017).…”

“…However, recently, Tanaka and Yoshioka (2017) found that this discretization becomes unstable even when the classical Courant-Friedrichs-Lewy (CFL) condition is satisfied. An implicit scheme was developed as an alternative to the semi-implicit one (Tanaka and Yoshioka, 2017). The semiimplicit discretization has been found to be physically inconsistent when the temporal resolution is not high, while the implicit one does not encounter this issue (Tanaka et al, 2018b).…”