Abstract Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid within Godunov-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.

中文翻译：

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（多）小波提高标准 Godunov 型流体动力学模型的准确性和效率：稳健的 2D 方法

摘要 多小波 (MW) 能够在基于二阶不连续伽辽金 (DG2) 和一阶有限体积 (FV1) 方法的 Godunov 型求解器内的多分辨率网格上压缩、分析和组装模型数据。已经使用一维 (1D) 流体动力学模型 (Kesserwani et al., 2019) 广泛研究了多小波自适应性，表明 MWDG2 可以比均匀 DG2 快 20 倍，比均匀 FV1 快 2 倍，同时保持准确性和稳健性基本公式。MWDG2 方案在二维 (2D) 建模方面的潜力尚未得到研究，但这需要一种能够稳健有效地求解具有复杂源项和润湿和干燥的二维浅水方程 (SWE) 的设计。本文提出了一个二维 MWDG2 方案：(1) 采用斜率解耦 DG2 求解器作为参考方案，因为它能够提供由简化的 3 分量基础形成的良好平衡的分段平面解；(2) 适应多小波的多分辨率分析以与斜率解耦 DG2 基兼容。斜率解耦 DG2 的缩放重构与两种多小波方法一起呈现，产生具有相似属性的 MWDG2 方案，以及用于适应分段常数模型数据的 Haar 小波 FV1 (HFV1) 变体。自适应 HFV1 和 MWDG2 求解器的性能与其统一求解器一起进行了探索，同时分析了它们的准确性、效率、网格粗化能力、处理跨越陡峭床坡的干湿锋的可靠性，以及捕捉与实际水力相关的特征的能力造型。结果表明一种特殊的多小波方法允许 MWDG2 方案利用其网格粗化能力来处理最广泛的流类型。结果还表明，所提出的基于（多）小波的自适应方案对于 2D 情况甚至更有效。随附的模型软件可在线公开获取。