Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green‐Naghdi equations

This paper presents a multiresolution discontinuous Galerkin scheme for the adaptive solution of Boussinesq‐type equations. The model combines multiwavelet‐based grid adaptation with a discontinuous Galerkin (DG) solver based on the system of fully nonlinear and weakly dispersive Green‐Naghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using multiwavelets on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way the local resolution level will be determined by manipulating multiwavelet coefficients controlled by a single user‐defined threshold value. The proposed adaptive multiwavelet discontinuous Galerkin solver for GN equations (MWDG‐GN) is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost

(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models

This paper presents a scaled reformulation of a robust second-order Discontinuous Galerkin (DG2) solver for the Shallow Water Equations (SWE), with guiding principles on how it can be naturally extended to fit into the multiresolution analysis of multiwavelets (MW). Multiresolution analysis applied to the flow and topography data enables the creation of an adaptive MWDG2 solution on a non-uniform grid. The multiresolution analysis also permits control of the adaptive model error by a single user-prescribed parameter. This results in an adaptive MWDG2 solver that can fully exploit the local (de)compression of piecewise-linear modelled data, and from which a first-order finite volume version (FV1) is directly obtainable based on the Haar wavelet (HFV1) for local (de)compression of piecewise-constant modelled data. The behaviour of the adaptive HFV1 and MWDG2 solvers is systematically studied on a number of well-known hydraulic tests that cover all elementary aspects relevant to accurate, efficient and robust modelling. The adaptive solvers are run starting from a baseline mesh with a single element, and their accuracy and efficiency are measured referring to standard FV1 and DG2 simulations on the uniform grid involving the finest resolution accessible by the adaptive solvers. Our findings reveal that the MWDG2 solver can achieve the same accuracy as the DG2 solver but with a greater efficiency than the FV1 solver due to the smoothness of its piecewise-linear basis, which enables more aggressive coarsening than with the piecewise-constant basis in the HFV1 solver. This suggests a great potential for the MWDG2 solver to efficiently handle the depth and breadth in resolution variability, while also being a multiresolution mesh generator. Accompanying model software and simulation data are openly available online

Benchmarking a multiresolution discontinuous Galerkin shallow water model: Implications for Computational hydraulics

Numerical modelling of wide ranges of different physical scales, which are involved in Shallow Water (SW) problems, has been a key challenge in computational hydraulics. Adaptive meshing techniques have been commonly coupled with numerical methods in an attempt to address this challenge. The combination of MultiWavelets (MW) with the Runge-Kutta Discontinuous Galerkin (RKDG) method offers a new philosophy to readily achieve mesh adaptivity driven by the local variability of the numerical solution, and without requiring more than one threshold value set by the user. However, the practical merits and implications of the MWRKDG, in terms of how far it contributes to address the key challenge above, are yet to be explored. This work systematically explores this, through the verification and validation of the MWRKDG for selected steady and transient benchmark tests, which involves the features of real SW problems. Our findings reveal a practical promise of the SW-MWRKDG solver, in terms of efficient and accurate mesh-adaptivity, but also suggest further improvement in the SWRKDG reference scheme to better intertwine with, and harness the prowess of, the MW-based adaptivity

Discontinuous Galerkin formulation for 2D hydrodynamic modelling: trade-offs between theoretical complexity and practical convenience

In the modelling of hydrodynamics, the Discontinuous Galerkin (DG) approach constitutes a more complex and modern alternative to the well-established finite volume method. The latter retains some desired practical features for modelling hydrodynamics, such as well-balancing between spatial flux and steep topography gradients, ability to incorporate wetting and drying processes, and computational affordability. In this context, DG methods were originally devised to solve the two-dimensional (2D) Shallow Water Equations (SWE) with irregular topographies and wetting and drying, albeit at reduction in the formulation’s complexity to often being second-order accurate (DG2). The aims of this paper are: (a) to outline a so-called “slope-decoupled” formulation of a standard 2D-DG2-SWE simulator in which theoretical complexity is deliberately reduced; (b) to highlight the capabilities of the proposed slopedecoupled simulator in providing a setting where the simplifying assumptions are verified within the formulation. Both the standard and the slope-decoupled 2D-DG2-SWE models adopt 2D modal basis functions for shaping local planar DG2 solutions on quadrilateral elements, by using an average coefficient and two slope coefficients along the Cartesian coordinates. Over a quadrilateral element, the stencil of the slope-decoupled 2D-DG2 formulation is simplified to remove the interdependence of slope-coefficients for both flow and topography approximations. The fully well-balanced character the slope-decoupled 2D-DG2-SWE planar solutions is theoretically studied. The performance of the latter is compared with the standard 2D-DG2 formulation in classical simulation tests. Other tests are conducted to diagnostically verify the conservative properties of the 2D-DG2-SWE method in scenarios involving sharp topography gradients and wet and/or dry zones. The analyses conducted offer strong evidence that the proposed slope-decoupled 2D-DG2-SWE simulator is very attractive for the development of robust flood models

Two-dimensional discontinuous Galerkin shallow water model for practical flood modelling applications

Finite volume (FV) numerical solvers to the two-dimensional shallow water equations are the foundation of the current state-of-the-practice, industry-standard flood models. The second-order Discontinuous Galerkin (DG2) alternative show a promising way to improve current FV-based flood model formulations, but is yet under-studied and rarely utilised to support flood modelling applications. This is contributed by the mathematical complexity constructed within the DG2 formulation that could lead to large computational costs and compromise its stability and robustness when used for practical modelling. Therefore, this PhD research aims to develop a new flood model based on simplified DG2 solver that is improved for flood modelling practices. To achieve this aim, three objectives have been formed and addressed through analyses involving academic and experimental test cases, as well as test cases that are recommended by the UK Environment Agency to validate 2D flood model capabilities, whilst benchmarking the simplified DG2 solver against four FV-based industrial models. Key research findings indicate that the simplified DG2 solver can equally retain conservative properties and provide second-order accurate predictions as the standard DG2 solver whilst offering around 2.6 times runtime speed up. Additionally, the simplified DG2 solver can be reliably efficient to provide predictions close to the outputs of the industrial models, in simulating flood scenarios covering large catchment-scale areas and at a grid resolution greater or equal than 5 m, particularly when the local limiting is disabled. However, the local limiting is still needed by the simplified DG2 solver when modelling detailed velocity fields at sub-metre grid resolutions, particularly in regions of highly active wave-structure interactions as commonly encountered in urban flooding around steep-sloped building structures