Multiwavelet-based mesh adaptivity with Discontinuous Galerkin schemes: Exploring 2D shallow water problems

(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models: Robust 2D approaches

“…Nonetheless, MWDG2 predictions at ε = 10 −2 can be deemed acceptable: beside the sources of uncertainty in this test (sensitivity to the choice of the Manning's coefficient and errors in experimental data measurement), its predictions are found to be in a better agreement with measured data compared with other predictions made by a second-order finite volume solver (FV2) on a triangular mesh type (Zhao et al, 2019), and by a third-order MWDG (MWDG3) solver designed with different treatments to achieve well-balancedness with wetting and drying (Caviedes-Voullième et al, 2020).…”

“…Although there is no unique choice for ε, a range of choices exists to keep the assembled DG2 solution on g A c (t) as accurate as the assembled solution on g (L) c (t). This optimal range for ε is expected to be somewhere between 10 −4 and 10 −2 within the scope of modelling shallow water flows (Kesserwani et al, 2019;Caviedes-Voullième et al, 2020), but is rather context-specific (Sharifian et al, 2019). An analysis of the choice for ε with the proposed adaptive HFV1 and MWDG2 schemes is carried out later in Section 3.1.1.…”

“…Kesserwani et al (2019) adopted multiwavelets and Haar wavelets to formulate adaptive one-dimensional (1D) MWDG2 and HFV1 hydrodynamic solvers, and found that the 1D-MWDG2 solver achieved the accuracy of the uniform DG2 solver for a runtime cost less than the uniform FV1 solver. The potential of (multi)wavelet-based adaptivity has yet to be explored for practical 2D hydrodynamic modelling, which requires the formulation of 2D-specific approaches for a robust 2D-MWDG2 scheme (Caviedes-Voullième and Caviedes-Voullième et al, 2020).…”

“…The first ingredient is a robust and efficient uniform 2D-DG2 solver serving as the underlying reference scheme. This 2D-DG2 solver must avoid spurious momentum errors at wet-dry fronts across very steep bed-slopes and reduce runtime cost (Caviedes-Voullième et al, 2020). Hence, the slope-decoupled 2D-DG2 solver (Kesserwani et al, 2018) is selected, in which piecewise-planar solutions are spanned by 3 scale coefficients-an average and two directionally independent slopes-on a simplified stencil.…”

Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods

“…It is based on a series of papers by Alpert, Belykin, and collaborators [1,9], and was introduced in the DG framework in [1]. It has since been used by Cheng, Gao, and collaborators for sparse representations [7,15,25,26,31], by Müller and collaborators for adaptivity [3,6,13], and by Vuik and Ryan for discontinuity detection [29,30]. It is based on the idea that a DG approximation over a mesh consisting of 2N elements, denoted u 2N h (x, t) , and be written as a DG approximation over a mesh consisting of N elements, denoted u N 2h (x, t) , together with the multi-wavelet information.…”