Numerical Treatment of the Resistance Term in Upwind Schemes in Debris Flow Runout Modeling

Abstract

Fast flows and avalanches of rock and debris are among the most dangerous of all landslide processes. Understanding and predicting postfailure motion (runout) of this kind of flowlike landslides is thus key for risk assessment, justifying the development of numerical models able to simulate their dynamics. In this paper a numerical method for the resolution of the depth-averaged debris flow model is presented. This set of nonlinear differential equations is formed by a variation of the shallow water equations, including strong bed slope, and a rheology resistance term. This paper focus on the numerical discretization of the resistance term, exploring three different approximations: pointwise, implicit, and unified. Well balance between numerical flux and source terms is only achieved using the unified discretization. In order to avoid nonphysical values of the water depth and discharge, a limitation of the unified resistance term is also needed. This correction is made following three conditions that identify the physical boundaries of the resistance term in the debris flow. This technique does not affect the computational efficiency of the method, keeping the original time step. Furthermore, proposed analytical test cases show that the three resistance limitations do not significantly perturb the numerical solution. The properties of the resulting numerical scheme are studied using a set of numerical experiments that include steady and transient flows. The results show the convenience of the unified discretization and the need of the three-condition limitation in order to avoid unphysical solutions.