Cellular Automata and Finite Volume solvers converge for 2D shallow flow modelling for hydrological modelling

Surface flows of hydrological interest, including overland flow, runoff, river and channel flow and flooding have received significant attention from modellers in the past 30 years. A growing effort to address these complex environmental problems is in place in the scientific community. Researchers have studied and favoured a plethora of techniques to approach this issue, ranging from very simple empirically-based mathematical models, to physically-based, deductive and very formal numerical integration of systems of partial-differential equations. In this work, we review two families of methods: cell-based simulators - later called Cellular Automata - and Finite Volume solvers for the Zero-Inertia equation, which we show to converge into a single methodology given appropriate choices. Furthermore, this convergence, mathematically shown in this work, can also be identified by critically reviewing the existing literature, which leads to the conclusion that two methods originating from different reasoning and fundamental philosophy, fundamentally converge into the same method. Moreover, acknowledging such convergence allows for some generalisation of properties of numerical schemes such as error behaviour and stability, which, importantly, is the same for the converging methodology, a fact with practical implications. Both the review of existing literature and reasoning in this work attempts to aid in the effort of synchronising and cross-fertilizing efforts to improve the understanding and the outlook of Zero-Inertia solvers for surface flows, as well as to help in clarifying the possible confusion and parallel developments that may arise from the use of different terminology originating from historical reasons. Moreover, synchronising and unifying this knowledge-base can help clarify model capabilities, applicability and modelling issues for hydrological modellers, specially for those not deeply familiar with the mathematical and numerical details.