abstract = {This PhD thesis aims to study the coupling of nonlinear shallow water models at different scales, with application to the numerical simulation of urban floods. Accurate simulations in this domain are usually prohibitively expensive due to the small mesh sizes necessary for the spatial discretization of the urban geometry and the associated small time steps constrained by stability conditions. Porosity-based shallow water models have been proposed in the past two decades as an alternative approach, consisting of upscaled models using larger mesh sizes and time steps and being able to provide good global approximations for the solution of the fine shallow water equations, with much smaller computational times. However, small-scale phenomena are not captured by this type of model. Therefore, we seek to formulate a numerical model coupling the fine and upscaled ones, in order to obtain more accurate solutions inside the urban zone, always with reduced computational costs relatively to the simulation of the fine model. The guideline for this objective lays on the use of predictor-corrector iterative parallel-in-time numerical methods, which naturally fit to this fine/coarse formulation. We focus on the parareal, one of the most popular parallel-in-time methods. As a main challenge, temporal parallelization suffers from instabilities and/or slow convergence when applied to hyperbolic or advection-dominated problems, such as the shallow water equations. Therefore, we consider a variant of the method using reduced-order models (ROMs) formulated on-the-fly along parareal iterations, using Proper Orthogonal Decomposition (POD) and the Empirical Interpolation Method (EIM), being able to improve the stability and convergence of the parareal method for solving nonlinear hyperbolic problems. We investigate the limitations of this ROM-based parareal method and we propose a number of modifications that provide further stability and convergence improvements: enrichment of the input snapshot sets used for the model reduction procedure; formulation of local-in-time ROMs; and incorporation of an adaptive parareal approach recently presented in the literature. The original and ROM-based parareal methods, including the proposed improvements, are compared and evaluated in terms of stability, convergence towards the fine solution and numerical speedup obtained in a parallel implementation. In a first part, the methods are formulated, studied and implemented considering a set of numerical simulations coupling the classical shallow water equations (without the porosity concept) at different scales. After this initial study, we implement them for coupling the classical and the porosity-based shallow water models, for the simulation of urban floods.},
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