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

Benchmarking (multi)wavelet-based dynamic and static non-uniform grid solvers for flood inundation modelling

This paper explores static non-uniform grid solvers that adapt three raster-based flood models on an optimised non-uniform grid: the second-order discontinuous Galerkin (DG2) model representing the modelled data as piecewise-planar fields, the first-order finite volume (FV1) model using piecewise-constant fields, and the local inertial (ACC) model only evolving piecewise-constant water depth fields. The optimised grid is generated by applying the multiresolution analysis (MRA) of multiwavelets (MWs) to piecewise-planar representation of raster-formatted topography data, for more sensible grid coarsening based on one user-specified parameter. Two adaptive solvers are also explored that apply the MRA of MWs and of Haar wavelets (HWs) to, respectively, scale and adapt the DG2 (MWDG2) and FV1 (HWFV1) modelled data dynamically in time. The performance of the non-uniform grid and adaptive solvers is assessed in terms of flood depth and extent, velocities, and CPU runtimes, with reference to the raster-based DG2 model predictions on their finest resolution grid. The assessments considered three large-scale flooding scenarios, involving rapid and slow-to-gradual flows. MWDG2 is found to be the most favourable choice when modelling rapid flows, where it excels in capturing small velocity variations. For slow-to-gradual flows, the adaptive solvers deliver less accurate outcomes, and their efficiency can be hampered by overhead costs of the dynamic MRA. Instead, non-uniform DG2 is recommended to capture urban flow interactions more accurately. Non-uniform ACC is 5 times faster to run than non-uniform DG2 but delivers close flooding depth and extent predictions, thus is more attractive for fluvial/pluvial flood simulation over large areas

Second-order discontinuous Galerkin flood model: comparison with industry-standard finite volume models

Finite volume (FV) numerical solvers of the two-dimensional shallow water equations are core to industry-standard flood models. The second-order Discontinuous Galerkin (DG) alternative is well-known to perform better than first- and second-order FV to capture sharp flow fronts and converge faster at coarser resolutions, but DG2 models typically rely on local slope limiting to selectively damp numerical oscillations in the vicinity of shock waves. Yet flood inundation events are smooth and gradually-varying, and shock waves play only a minor role in flood inundation modelling. Therefore, this paper investigates two DG2 variants - with and without local slope limiting - to identify the simplest and most efficient DG2 configuration suitable for flood inundation modelling. The predictive capabilities of the DG2 variants are analysed for a synthetic test case involving advancing and receding waves representative of flood-like flow. The DG2 variants are then benchmarked against industry-standard FV models over six UK Environment Agency scenarios. Results indicate that the DG2 variant without local slope limiting closely reproduces solutions of the commercial models at twice as coarse a spatial resolution, and removing the slope limiter can halve model runtime. Results also indicate that DG2 can capture more accurate hydrographs incorporating small-scale transients over long-range simulations, even when hydrographs are measured far away from the flooding source. Accompanying details of software and data accessibility are provided

A discontinuous Galerkin approach for conservative modelling of fully nonlinear and weakly dispersive wave transformations

This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green-Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modelling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model

Simulation of laminar to transitional wakes past cylinders with a discontinuous Galerkin inviscid shallow water model

Laminar to transitional wakes occur in slow, quasi-steady flows past cylinders at low cylinder Reynolds numbers (R ed ≤ 250). Inviscid numerical solvers of the depth-averaged shallow water equations (SWE) introduce numerical dissipation that, depending on Red, may imitate the mechanisms of viscous turbulent models. However, the numerical dissipation rate in a second-order finite volume (FV2) SWE solver is so large at a practical resolution that this can instead hide these mechanisms. The extra numerical complexity of the second-order discontinuous Galerkin (DG2) SWE solver results in a lower dissipation rate, making it a potential alternative to the FV2 solver to reproduce cylinder wakes. This paper compares the DG2 and FV2 solvers, initially for wake formation behind one cylinder. The findings confirm that DG2 can reproduce the expected wake formations, which FV2 fails to capture, even at a 10-fold finer resolution. It is further demonstrated that DG2 is capable of reproducing key features of the flow fields observed in a laboratory random cylinder array

LISFLOOD-FP 8.1: new GPU-accelerated solvers for faster fluvial/pluvial flood simulations

The local inertial two-dimensional (2D) flow model on LISFLOOD-FP, the so-called ACCeleration (ACC) uniform grid solver, has been widely used to support fast, computationally efficient fluvial/pluvial flood simulations. This paper describes new releases, on LISFLOOD-FP 8.1, for parallelised flood simulations on the graphical processing units (GPUs) to boost efficiency of the existing parallelised ACC solver on the central processing units (CPUs) and enhance it further by enabling a new non-uniform grid version. The non-uniform solver generates its grid using the multiresolution analysis (MRA) of the multiwavelets (MWs) to a Galerkin polynomial projection of the digital elevation model (DEM). This sensibly coarsens the resolutions where the local topographic details are below an error threshold ε and allows classes of land use to be properly adapted. Both the grid generator and the adapted ACC solver on the non-uniform grid are implemented in a GPU new codebase, using the indexing of Z-order curves alongside a parallel tree traversal approach. The efficiency performance of the GPU parallelised uniform and non-uniform grid solvers is assessed for five case studies, where the accuracy of the latter is explored for and 10−3 in terms of how close it can reproduce the prediction of the former. On the GPU, the uniform ACC solver is found to be 2–28 times faster than the CPU predecessor with increased number of elements on the grid, and the non-uniform solver can further increase the speed up to 320 times with increased reduction in the grid's elements and decreased variability in the resolution. LISFLOOD-FP 8.1, therefore, allows faster flood inundation modelling to be performed at both urban and catchment scales. It is openly available under the GPL v3 license, with additional documentation at https://www.seamlesswave.com/LISFLOOD8.0 (last access: 12 March 2023)

LISFLOOD-FP 8.0 : the new discontinuous Galerkin shallow-water solver for multi-core CPUs and GPUs

LISFLOOD-FP 8.0 includes second-order discontinuous Galerkin (DG2) and first-order finite-volume (FV1) solvers of the two-dimensional shallow-water equations for modelling a wide range of flows, including rapidly propagating, supercritical flows, shock waves or flows over very smooth surfaces. The solvers are parallelised on multi-core CPU and Nvidia GPU architectures and run existing LISFLOOD-FP modelling scenarios without modification. These new, fully two-dimensional solvers are available alongside the existing local inertia solver (called ACC), which is optimised for multi-core CPUs and integrates with the LISFLOOD-FP sub-grid channel model. The predictive capabilities and computational scalability of the new DG2 and FV1 solvers are studied for two Environment Agency benchmark tests and a real-world fluvial flood simulation driven by rainfall across a 2500 km2 catchment. DG2's second-order-accurate, piecewise-planar representation of topography and flow variables enables predictions on coarse grids that are competitive with FV1 and ACC predictions on 2–4 times finer grids, particularly where river channels are wider than half the grid spacing. Despite the simplified formulation of the local inertia solver, ACC is shown to be spatially second-order-accurate and yields predictions that are close to DG2. The DG2-CPU and FV1-CPU solvers achieve near-optimal scalability up to 16 CPU cores and achieve greater efficiency on grids with fewer than 0.1 million elements. The DG2-GPU and FV1-GPU solvers are most efficient on grids with more than 1 million elements, where the GPU solvers are 2.5–4 times faster than the corresponding 16-core CPU solvers. LISFLOOD-FP 8.0 therefore marks a new step towards operational DG2 flood inundation modelling at the catchment scale. LISFLOOD-FP 8.0 is freely available under the GPL v3 license, with additional documentation and case studies at https://www.seamlesswave.com/LISFLOOD8.0 (last access: 2 June 2021)

Survey and feasibility study for the introduction of native fish and non- native fish for cage culture in the southern part of Caspian Sea

The aim of this study is the feasibility of introducing suitable species of fish (native and exotic) for rearing in cages in the southern region of the Caspian Sea. This study from the perspective of economic efficiency and maintaining ecology were analyzed. The results showed that the southern Caspian Sea has the ability to aquaculture fish. However, due to the yearly temperature variations of water (from surface to a depth of 50 meters), the talent of aquaculture area is preferred for fish in cold water than warm water fish due to possibility rearing over the years. The survey showed that the current conditions and due to unfavorable changes in the ecology of the area in the last decade, the use of non-native fish culture in cages is not recommended. Many species of native fish are suitable for rearing in cages. Nevertheless, there is no infrastructure suitable for the production of all of them. Therefore, respectively species of Salmo caspius, Huso huso and Common carp of Caspian Sea for cultivation of in cages was suggested. In the current situation, this fish for their ability to grow faster than the need for selectivity. It is noteworthy that detailed environmental assessments and species risk assessments before the final introduction of any fish species (native and exotic) for rearing in cages in the Caspian Sea ecosystem is essential

(Multi)wavelet-based Godunov-type simulators of flood inundation: Static versus dynamic adaptivity

Real-world flood simulators often use first-order finite volume (FV1) solvers of the shallow water equations with efficiency enhancements exploiting parallelisation on Graphical Processing Units (GPUs) and the use of static adaptivity on fixed grids. A second-order discontinuous Galerkin (DG2) solver greatly increases the accuracy in the predictions on uniform grids, where it is comparatively costly to run, but its practical utility as an alternative for flood simulations using static adaptivity is not yet assessed. This is also the case for the dynamic adaptivity using the multiresolution analysis (MRA) of the Haar wavelet (HW) scaling FV1 piecewise-constant solutions (HWFV1) and of the smoother Multiwavelets (MWs) that scales DG2 piecewise-planar solutions (MWDG2) to adapt the resolution of their grids over time. Therefore, dynamic MWDG2 and HWFV1 adaptivity is newly explored for practical real-world simulations, to find out when they yield better predictions than static DG2 and FV1 adaptivity. A new GPU implementation is proposed to include dynamic MWDG2 adaptivity to also assess how far GPU parallelisation renders its runtime practically feasible. Dynamic and static adaptivity are assessed for three tests involving slow, gradual to rapid flood flows with analyses of their predictive accuracy and computational costs with reference to uniform grid DG2 simulations at the finest resolution of the digital elevation model (DEM). Findings suggest favouring static FV1 adaptivity for long-duration simulations of slowly to gradually propagating floods and dynamic MWDG2 adaptivity to simulate events driven by rapidly propagating flows. On the GPU, dynamic MWDG2 adaptivity is faster than uniform DG2, leading to a higher speedup ratio with higher reduction in the elements on its initial, fixed grid

Adaptive multi-scale shallow flow model: a wavelet-based formulation

This work outlines the use of wavelet bases to re-formulate a finite volume (FV) local solution of the shallow water equations (SWEs), so as to achieve mesh adaptivity via local compression and truncation of the numerical solution’s details across successive resolution scales with reference to a single threshold error set by the user. The wavelet bases naturally lead to a scalable FV formulation and how they can readily be exploited to achieve adaptive mesh-resolution selection: up-scaling and/or down- scaling by means of the local solutions’ data (i.e. both flow variables and terrain). Our results show a notable promise in using wavelets as a basis for future flood models to achieve conservative and more autonomous simulation at a wide range of length-scales