GraphFlood: a fast stationary solution for 2D hydrodynamics in landscapes evolution models
Authors: Gailleton Boris , Steer Philippe , Davy Philippe , Schwanghart Wolfgang , Bernard Thomas 
Published: February 2023

Integrating hydrology in Landscapes Evolution Models (LEMs) is challenging. The drainage-area-based solutions, where drainage area weighted by precipitation rates approximate the amount of water flowing through every location, accumulates drainage-area downstream following the topographic gradient and has been empirically linked to observed discharge. While straightforward and computationally efficient, it implicitly includes hydrology without calculating discharge or water height. A more sophisticated solution consist in the direct calculation of the shallow water equation, which explicitly approximates water height and discharge using physics-based equations. While the latter bears more information about the channels and floodplains dynamics, it is inherently limited to short time scales and is computationally more expensive, with numerical time step typically of the order of the second &#8211; making its use for long-term LEMs particularly challenging. Here, we present GraphFlood, a fast iterative method computing river depth and water discharge in 2D on a digital elevation model (DEM). This new method leverages the Directed Acyclic Graph (DAG) nature of water flowing on surface topography to iteratively solve for the 2D shallow water equation without the inertia terms. The main idea of the algorithm is to find the correct water surface height by iteratively balancing discharge input and output. At each iteration, we first use fast DAGs-related algorithms to calculate flow accumulation on the hydraulic surface. We use this as an approximation of the discharge input. Then, the discharge output is calculated using the Manning flow resistance equation, in a manner similar to the Floodos model (Davy et al., 2017). The divergence of the discharges increments the water height. The iterative process is repeated until reaching a stationary state, i.e. a static field of water height and discharge representing an equilibrium state. Note that this method can be slightly modified to solve flood wave propagation by approximating the input discharge function of the immediate upstream neighbours. Water depths obtained with the stationary solution were validated against an analytical solution in the case of a rectangular channel and with the Floodos model for natural DEMs. Compared to previous hydrodynamic models, the main benefits of GraphFlood are its simplicity of implementation, which mainly requires a classical flow routing algorithm, and its efficiency. While case-dependent, our tests suggested a ~10 times speed-up compared to Floodos model (Davy et al., 2017) which was already significantly faster than other hydrodynamic models. Moreover, the computational time scales a little more than linearly with the number of cells, which makes GraphFlood a suitable solution even for DEMs larger than 106 &#8211; 107 cells. We demonstrate the suitability of the method for integrating realistic hydrology in a wide range of topographic and morphometric analyses (e.g. channel width assessment, floodplain delineation, Flint-Morisawa metrics) and in LEMs &#8211; even for longer timescales.&#160;