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

Abstract

Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid withinGodunov-type solvers based on second-order discontinuous Galerkin (DG2) and first-order finite volume (FV1) methods.Multiwavelet adaptivity has been studied extensively with one-dimensional (1D) hydrodynamic models (Kesserwani et al., 2019), revealing that MWDG2 can be 20 times faster than uniform DG2 and 2 times faster than uniform FV1, while preserving the accuracy and robustness of the underlying formulation. The potential of the MWDG2 scheme has yet to be studied for two-dimensional (2D) modelling, but this requires a design that robustly and efficiently solves the 2D shallow water equations (SWE) with complex source terms and wetting and drying. This paper presents a two-dimensional MWDG2 scheme that: (1) adopts a slope-decoupled DG2 solver as a reference scheme, for its ability to deliver well-balanced piecewise-planar solutions shaped by a simplified 3-component basis; and, (2) adapts the multiresolution analysis of multiwavelets for compatibility with the slope-decoupled DG2 basis. A scaled reformulation of slope-decoupled DG2 is presented alongside two multiwavelet approaches that yield MWDG2 schemes with similar properties, and a Haar wavelet FV1 (HFV1) variant for adapting piecewise-constant model data. The performance of the adaptive HFV1 and MWDG2 solvers is explored alongside their uniform counterparts, while analysing their accuracy, efficiency, grid-coarsening ability, reliability in handling wet-dry fronts across steep bed-slopes, and ability to capture features relevant to practical hydraulic modelling. The results indicate a particular multiwavelet approach that allows the MWDG2 scheme to exploit its grid-coarsening ability for the widest range of flow types. Results also indicate that the proposed (multi)wavelet-based adaptive schemes are even more efficient for the 2D case. Accompanying model software is openly available online.

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

“…Within the element the discrete flow vector U h (x, y) and topography z h (x, y) are represented by locally planar fields. Expressed as a scaled Legendre basis expansion (Kesserwani and Sharifian, 2020), the flow vector U h (x, y) is written as…”

“…While this friction scheme has been successfully adopted in finite-volume and discontinuous Galerkin settings for modelling dam break flows and urban flood events (Wang et al, 2011;Kesserwani and Wang, 2014), it can exhibit spuriously large velocities and correspondingly small time steps for large-scale, rainfall-induced overland flows, involving widespread, very thin water layers flowing down hill slopes and over steep riverbanks, as demonstrated by Xia et al (2017). Due to the involvement of the slope coefficients, water depths at Gaussian quadrature points can be much smaller (and velocities much larger) than the element-average values.…”

“…However, this approach relies on the availability of observation data, and, due to modelling sensitivities at the scale of the grid, optimal positions can vary depending on the choice of solver and grid resolution. Spatially adaptive solvers (Kesserwani and Sharifian, 2020;Özgen-Xian et al, 2020) and non-uniform meshing techniques (Kolega and Syme, 2019) offer another alternative to improve flow predictions by selectively capturing fine-scale channel geometries, and such methods are under development for inclusion in a future LISFLOOD-FP release. Sub-grid channel modelling can also improve hydrograph and flood inundation predictions, and LISFLOOD-FP already provides a sub-grid channel model (Neal et al, 2012a) that could be integrated with the DG2 and FV1 solvers in a future release.…”

“…Within a computational element on a raster grid, each locally planar variable is represented by three coefficients -the element-average, x-slope and y-slope coefficients -which are updated by a two-stage Runge-Kutta timestepping scheme. Due to its second-order formulation, DG2 can be 4-12 times slower per element than an FV1 solver depending on the test case (Kesserwani and Sharifian, 2020), though substantial speed-ups have already been achieved: switching from a standard tensor-product stencil to a simplified, slope-decoupled stencil of Kesserwani et al (2018) achieved a 2.6-fold speed-up, and avoiding unnecessary local slope limiting achieved an additional 2-fold speed-up (Ayog et al, 2021), while preserving accuracy, conservation and robustness properties for shockless flows.…”

“…Second-order finite-volume (FV2) methods offer an alternative approach to obtain second-order accuracy, with many FV2 models adopting the Monotonic Upstream-centred Scheme for Conservation Laws (MUSCL) method. While FV2-MUSCL solvers can achieve second-order convergence (Kesserwani and Wang, 2014), the MUSCL method relies on global slope limiting and non-local, linear reconstructions across neighbouring elements that can affect energy conservation properties (Ayog et al, 2021) and affect wave arrival times when the grid is too coarse (Kesserwani and Wang, 2014). Hence, although FV2-MUSCL is typically 2-10 times faster than DG2 per element (Ayog et al, 2021), DG2 can improve accuracy and conservation properties on coarse grids, which is particularly desirable for efficient, long-duration continental-or global-scale simulations that rely on DEM products derived from satellite data (Bates, 2012;Yamazaki et al, 2019).…”