GPU-parallelisation of Haar wavelet-based grid resolution adaptation for fast finite volume modelling: application to shallow water flows

Abstract

Wavelet-based grid adaptation driven by the ‘multiresolution analysis’ (MRA) of the Haar wavelet (HW) allows to devise an adaptive first-order finite volume (FV1) model (HWFV1) that can readily preserve the modelling fidelity of its reference uniform-grid FV1 counterpart. However, the MRA incurs a high computational cost as it involves ‘encoding’ (coarsening), ‘decoding’ (refining), analysing and traversing modelled data across a deep hierarchy of nested, uniform grids. GPU-parallelisation of the MRA is needed to reduce its computational cost, but its algorithmic structure (1) hinders coalesced memory access on the GPU and (2) involves an inherently sequential tree traversal problem. This work redesigns the algorithmic structure of the MRA in order to parallelise it on the GPU, addressing (1) by applying Z-order space-filling curves and (2) by adopting a parallel tree traversal algorithm. This results in a GPU-parallelised HWFV1 model (GPU-HWFV1). GPU-HWFV1 is verified against its CPU predecessor (CPU-HWFV1) and its GPU-parallelised reference uniform-grid counterpart (GPU-FV1) over five shallow water flow test cases. GPU-HWFV1 preserves the modelling fidelity of GPU-FV1 while being up to 30 times faster. Compared to CPU-HWFV1, it is up to 200 times faster, suggesting that the GPU-parallelised MRA could be used to speed up other FV1 models.

GPU‐Accelerated Urban Flood Modeling Using a Nonuniform Structured Grid and a Super Grid Scale River Channel

“…We briefly summarize the steps to generate a locally adapted grid that corresponds to the underlying topography features. While most AMR methods create a nonuniform grid by selectively refining a baseline grid at the coarsest resolution, the wavelet‐based multiresolution AMR approach diverges from this by initiating with a baseline uniform grid at the finest resolution and coarsens it (Chowdhury et al., 2023; Sharifian et al., 2023). Like data compression in image processing, wavelet‐based AMR can be achieved in three steps, namely multiscale decomposition, hard thresholding, and grid adaptation:…”

“…Overall accuracies are captured within a given tolerance ε while extracting the inherent complexity of the problem with as few degrees of freedom as possible. A depth‐first traversal algorithm from coarse levels to fine is then performed to collect all the leaf elements, thereby forming the nonuniform grid (Chowdhury et al., 2023; Sedgewick & Wayne, 2011).…”

“…Crucial grid details, like neighbour references and parent‐child relationships, are stored in a hierarchical data structure, facilitating easy retrieval for grid adjustments (Hou et al., 2018; Liang, 2011). Yet, the memory arrangement and the cost of accessing graph‐like data structures, as opposed to plain matrix memorization, cannot exploit the computational capability of GPUs, given that optimal GPU throughput is achieved through regularity in computer memory accesses (Chowdhury et al., 2023; Vacondio et al., 2017). It is noteworthy that the core of AMR lies in designing refinement criteria based on interpolation error estimates of some key quantity (e.g., water surface gradient, velocity).…”

“…Thus, the key contribution in this paper is the integration of a novel 1D channel representation with a nonuniform structured 2D floodplain hydrodynamic model and then coding this for application on a GPU. Previous nonuniform structured approaches in hydrodynamics (Chowdhury et al., 2023; Kesserwani & Sharifian, 2023; Sharifian et al., 2023) have been, to the best of our knowledge, pure 2D only, but this is computationally inefficient for representing riverine urban flooding where we are not concerned with fine‐scale details of in‐channel flow. Instead, for practical flood simulations hybrid 1D‐2D models are preferred, but to date such models have used uniform 2D grids for the floodplain.…”