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In recent years the concept of multiresolution-based adaptive discontinuous Galerkin (DG) schemes for hyperbolic conservation laws has been developed. The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defined on a hierarchy of nested grids. Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element. For non-uniform grid hierarchies multiwavelets have to be constructed for each element and, thus, becomes extremely expensive. To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

<p><span>Shallow water modelling is a widely used for a vast range of applications in Hydraulics, Hydrology and Environmental Geosciences. It is at the core of most fluvial flood modelling approaches, and increasingly turning into the model of choice for urban flood modelling, coastal modelling and rainfall-runoff hydrological simulation. Shallow water solvers have significantly matured in the last decade, and currently, robust and accurate first-order solvers are widely available. Relevant developments have also been achieved in terms of higher order solvers, based on MUSCL and WENO reconstructions and on Discontinuous Galerkin (DG) schemes. Despite all this, applying shallow water solvers on realistic problems is constrained by the multiscale nature of environmental surface flows, in which flows in large domains are strongly affected by small-scale features of both the topography and the flow fields. This inherently multiscale problem naturally calls for a multiresolution modelling strategy, which is the topic of this contribution.</span></p><p> </p><p><span>In this work, we explore the application of a multidimensional Discontinuous Galerkin scheme with dynamic mesh adaptivity driven by multiresolution analysis based on wavelets. The scheme harnesses the locality and high-order properties of DG, and makes use of an additional decomposition into the multiwavelet space driving a multiresolution analysis. By assessing the relevance of local features of the solution across scales, mesh adaptivity is triggered. In previous works, the general scheme has been presented and tested. Herein, we test the capabilities of the scheme on well-known benchmark problems for 2D shallow flows, including both laboratory and field scale flows. </span></p><p> </p><p><span>The results clearly show that the scheme is capable of solving such problems with a high accuracy and that the dynamically adaptive mesh is capable of tracking physically-meaningful interfaces (wetting and drying fronts, transcritical shocks, rotating vortices) accurately. Moreover, the adaptive scheme is capable of providing very high spatial resolution where and when it is required, while keeping the computational cost orders of magnitude lower than what a uniform high resolution mesh would impose. In particular, the results suggest that this type of adaptive scheme produces more efficient meshes than alternative schemes. The results showcase some of the advantages of high-order solvers, especially when combined with adaptive schemes and are a proof-of-concept of the applicability of this type of solvers for realistic problems. Finally, the results also evidence the capability of the adaptive multiresolution strategy to transparently incorporate the properties of the underlying shallow water solver, allowing for improvements on the core scheme to always benefit the adaptive solution.</span></p>