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Sustainable urban drainage systems are multi-functional nature-based solutions that can facilitate flood management in urban catchments while improving stormwater runoff quality. Traditionally, the evaluation of the performance of sustainable drainage infrastructure has been limited to a narrow set of design objectives to simplify their implementation and decision-making process. In this study, the spatial design of sustainable urban drainage systems is optimized considering five objective functions, including minimization of flood volume, flood duration, average peak runoff, total suspended solids, and capital cost. This allows selecting an ensemble of admissible portfolios that best trade-off capital costs and the other important urban drainage services. The impact of the average surface slope of the urban catchment on the optimal design solutions is discussed in terms of spatial distribution of sustainable drainage types. Results show that different subcatchment slopes result in non-uniform distributional designs of sustainable urban drainage systems, with higher capital costs and larger surface areas of green assets associated with steeper slopes. This has two implications. First, urban areas with different surface slopes should not have a one-size-fits-all design policy. Second, spatial equality must be taken into account when applying optimization models to urban subcatchments with different surface slopes to avoid unequal distribution of environmental and human health co-benefits associated with green drainage infrastructure.

Flux vectors' matrix h Water depth h e Element size h L Water depth in the upstream channel h R Water depth in the downstream channel h i Numerical depth results in each node i Node number K Stiffness matrix L Length of the computational domain M Mass matrix n Manning's roughness coefficient S Topographical and frictional source terms S bx Depth gradients in the x-directions S by Depth gradients in the y-directions S fx Friction slopes along x-directions S fy Friction slopes along y-directions s Bore speed t Time U Matrix of variables u Depth-integrated velocity in the x-directions v Depth-integrated velocity in the y-directions x x-direction space y y-direction space Ψ i Basis-function Abstract A new numerical scheme based on the finiteelement method with a total-variation-diminishing property is developed with the aim of studying the shock-capturing capability of the combination. The proposed model is formulated within the framework of the one-step Taylor-Galerkin scheme in conjunction with the Van Leer's limiter function. The approach is applied to the two-dimensional shallow water equations by different test cases, i.e., the partial, circular, and one-dimensional dam-break flow problems. For the one-dimensional case, the sub-and supercritical flow regimes are considered. The results are compared with the analytical, finite-difference, and smoothed particle hydrodynamics solutions in the literature. The findings show that the proposed model can effectively mask the sources of errors in the abrupt changes of the flow conditions and is able to resolve the shock and rarefaction waves where other numerical models produce spurious oscillations.