Abstract. We develop a network-based model of a catchment basin that incorporates the possibility of small-scale, in-channel, leaky barriers as flood attenuation features, on each of the edges of the network. The model can be used to understand effective risk reduction strategies considering the whole-system performance; here we focus on identifying network dam placements promoting effective dynamic utilisation of storage and placements that also reduce risk of breach or cascade failure of dams during high flows. We first demonstrate the model using idealised networks and explore risk of cascade failure using probabilistic barrier-fragility assumptions. The investigation highlights the need for robust design of nature-based measures, to avoid inadvertent exposure of communities to a flood risk, and we conclude that the principle of building the leaky barriers on the upstream tributaries is generally less risky than building on the main trunk, although this may depend on the network structure specific to the catchment under study. The efficient scheme permits rapid assessment of the whole-system performance of dams placed in different locations in real networks, demonstrated in application to a real system of leaky barriers built in Penny Gill, a stream in the West Cumbria region of Britain.
Abstract. We develop a network-based model of a catchment basin that incorporates the possibility of small-scale, in-channel, leaky barriers as flood attenuation features, on each of the edges of the network. The model can be used to understand effective risk reduction strategies considering the whole-system performance; here we focus on identifying network dam placements promoting effective dynamic utilisation of storage, and placements that also reduce risk of breach or cascade failure of dams during high flows. We first demonstrate the model using idealised networks and explore risk of cascade failure using probabilistic barrier-fragility assumptions. The investigation highlights the need for robust design of nature-based measures, to avoid inadvertent exposure of communities to a flood risk, and we conclude that the principle of building the leaky-barriers on the upstream tributaries is generally less risky than building on the main trunk, although this may depend on the network structure specific to the catchment under study. The efficient scheme permits rapid assessment of performance of dams placed in different locations in real networks, demonstrated in application to a real system of leaky barriers built in Penny Gill, a stream in the West Cumbria region of Britain and which leads to further design advice.
Efficient parallel-in-time methods for hyperbolic partial differential equation problems remain scarce. Here we investigate an approach based on circulant preconditioned generalised minimal residual (GMRES) for the monolithic block Toeplitz equations which arise from constant time-step discretizations. We present theoretical results which guarantee convergence in a number of iterations independent of the number of time-steps and demonstrate the potential utility of the approach with numerical results employing several different finite difference schemes of varying orders of accuracy.
Two of the most popular parallel-in-time methods are Parareal and multigrid-reduction-in-time (MGRIT). Recently, a general convergence theory was developed in Southworth (2019) for linear two-level MGRIT/Parareal that provides necessary and sufficient conditions for convergence, with tight bounds on worst-case convergence factors. This paper starts by providing a new and simplified analysis of linear error and residual propagation of Parareal, wherein the norm of error or residual propagation is given by one over the minimum singular value of a certain block bidiagonal operator. New discussion is then provided on the resulting necessary and sufficient conditions for convergence that arise by appealing to block Toeplitz theory as in Southworth (2019). Practical applications of the theory are discussed, and the convergence bounds demonstrated to predict convergence in practice to high accuracy on two standard linear hyperbolic PDEs: the advection(-diffusion) equation, and the wave equation in first-order form.
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