## 1. Introduction

[2] The past few decades have witnessed exceptional progress in the development of computer models to simulate the propagation of floods, much of which has been driven by the need for tools to evaluate flood risk. Most of these methods are based on the shallow water equations, a nonlinear hyperbolic system of equations describing the conservation of mass and momentum of water in two horizontal dimensions. In spite of the substantial progress achieved in key areas of numerical methods for the solution of this system and the unprecedented evolution of computational power, computations over large space and time scales and at high resolution are still hampered by the unfeasibly high computational cost involved. With the growing availability of fine resolution remotely sensed data, this computational limitation represents a clear barrier for a range of applications aimed at understanding and predicting large-scale flood problems.

[3] The demand for computationally efficient methodologies to solve flood propagation problems has led to the development of simplified formulations that provide accurate solutions within the range of their applicability at a considerably high performance. The two most popular examples are the kinematic and diffusion wave approximations. The first is derived by neglecting the two Eulerian inertial terms in the momentum equation (i.e., local and convective acceleration) along with the pressure gradient term, while the second neglects inertia terms only. The properties and limitations of these models—as well as of other models based on simplifications of the Saint Venant equations (i.e., quasi-steady dynamic wave and gravity wave)—have been widely explored over the last decades [e.g., *Lighthill and Whitham*, 1955; *Woolhiser and Liggett*, 1967; *Di Silvio*, 1969; *Ponce and Simons*, 1977; *Ponce et al*., 1978; *Morris and Woolhiser*, 1980; *Vieira*, 1983; *Ferrick et al*., 1984; *Ferrick*, 1985; *Ponce*, 1986; *Hromadka and Yen*, 1986; *Dooge and Napiorkowski*, 1987; *Govindaraju et al*., 1988a, 1988b, 1990; *Parlange et al*., 1990; *Singh and Aravamuthan*, 1996; *Lamberti and Pilati*, 1996; *Cappelaere*, 1997; *Tsai*, 2003; *Tsai and Yen*, 2004; *Moramarco et al*., 2008; *Philipp et al*., 2012]. Much less attention has been given to a third class of shallow water approximate solvers—the so-called local inertial approximation—which neglects only the convective part of the inertial terms [e.g., *Xia*, 1995, *Aronica et al*., 1998, *Bates et al*., 2010, *de Almeida et al*., 2012].

[4] From a conceptual point of view, the local inertial formulation provides a better physical description of the problem than the diffusive model, as the local rate of change of the fluid momentum is now represented by the additional term. In discrete terms, this means that the fluid momentum at any given time step is used to update the next time step, so that the flow has to be accelerated from a previous state. Therefore, in terms of the physical representation of shallow water flows, the local inertial formulation lies between the diffusion wave approximation and the full-dynamic equations. From the computational perspective, the main advantage of the local inertial formulation (compared to the diffusion wave model) is related to the stability condition in explicit finite difference schemes. Namely, the maximum stable time step in the explicit diffusion wave model decreases quadratically with grid refinements [see *Hunter et al*., 2006], while a linear relation is obtained with the numerical schemes based on the solution of the hyperbolic local inertial system (which is governed by the Courant-Friedrichs-Lewy condition, hereafter referred to as CFL). This key difference substantially enhances the computational performance at fine resolution. In addition, the stable time step for the diffusion wave model also depends on the water surface gradient [see *Hunter et al*., 2006], so that its computational performance is dramatically reduced in zones of near horizontal water surface. This is particularly important for large-scale simulations that unavoidably include flat water surface areas (e.g., lakes and lowland rivers). The local inertial model therefore provides an excellent alternative for the simulation of flow problems that would become unfeasibly slow with an explicit diffusive model.

[5] *Bates et al*. [2010] proposed a simple finite difference numerical scheme for the solution of the local inertial system that provides accurate solutions to a range of subcritical flow conditions at a very attractive computational performance. A number of subsequent studies have benchmarked this model against field data, analytical solutions, as well as the solutions of a range of industry numerical models that solve the full-dynamic and simplified versions of the shallow water equations [e.g., *Bates et al*., 2010, *Neal et al*., 2012, *Falter et al*., 2012]. Recently, an alternative numerical scheme was proposed by *de Almeida et al*. [2012] that solved some stability problems reported by *Bates et al*. [2010], providing a robust and efficient model to simulate a range of subcritical flood propagation problems.

[6] *Neal et al*. [2012] compared the performance of the *Bates et al*. [2010] local inertial model against a diffusion wave and a full-dynamic model (a Godunov-type finite volume scheme that uses the approximate Riemann solver of Roe). They observed that the local inertial model performed up to seven times faster than the full-dynamic model, and more than 2 orders of magnitude faster than the diffusion wave model. The speed-up of model performance compared to a full-dynamic model obviously comes at the cost of neglecting one of the terms in the governing equations, and it is the objective of this paper to analyze how this assumption affects the accuracy of the model.

[7] The relative simplicity of the resulting set of algebraic finite difference equations [e.g., *Bates et al*., 2010; *de Almeida et al*., 2012] not only reduces the computational cost but also significantly increases its potential to be implemented as part of a range of models that would benefit from a simple, robust, and efficient two-dimensional (2-D) hydraulic module. This is reflected in recent efforts to implement the *Bates et al*. [2010] scheme as the hydrodynamic engine of the CAESAR landscape evolution model (e.g., Coulthard et al., Integrating the LISFLOOD-FP 2-D hydrodynamic model with the CAESAR model: Implications for modelling landscape evolution, submitted to *Earth Surface Processes and Landforms*, 2013) and JULES, the UK Land Environment Simulator.

[8] With an increasing interest in this formulation, it becomes of paramount importance to understand its main capabilities and limits of applicability to simulate flood propagation problems. This paper is organized as follows. First, the governing equations and the corresponding numerical scheme used to solve the system are briefly presented. This is followed by a discussion comparing the main differences between the full-dynamic and local inertial systems, which reveal some of the properties of the latter with a potential influence on practical problems. The analysis is structured around two main aspects of modeling capabilities: (i) the ability of the model to accurately simulate steady flow problems and (ii) flood propagation (one-dimensional (1-D) unsteady flow). The mathematical insights developed in these sections are supported by and further analyzed through a sequence of test cases with analytical solutions to the full-dynamic equations. These test cases are used to illustrate the most relevant differences in the behavior of the solutions obtained with the two systems. The tests were specifically designed to explore the same aspects of the local inertial model previously discussed and cover a wide range of flow conditions. Finally, the results of the local inertial and full-dynamic models are discussed in light of the properties of the two systems. These results illustrate the accuracy obtained with the local inertial model at a range of subcritical flow conditions, as well as the main flow characteristics influencing this accuracy.