We develop a variant of a tensor reduced-order model (tROM) for the parameterized shallow-water dam-break problem. This hyperbolic system presents multiple challenges for model reduction, including a slow decay of the Kolmogorov $N$-width of the solution manifold, shock formation, and the loss of smooth solution dependence on parameters. These issues limit the performance of traditional Proper Orthogonal Decomposition based ROMs. Our tROM approach, based on a low-rank tensor decomposition, builds a parameter-to-solution map from high-fidelity snapshots and constructs localized reduced bases via a local POD procedure. We apply this method to both dry-bed and wet-bed problem setups, showing that the non-interpolatory variant of the tROM, combined with Chebyshev sampling near critical parameter values, effectively captures parameter-dependent behavior and significantly outperforms standard POD-ROMs. This is especially evident in the wet-bed case, where POD-ROMs exhibit poor resolution of shock waves and spurious oscillations.

翻译：我们针对参数化浅水溃坝问题开发了一种张量降阶模型（tROM）的变体。该双曲系统对模型降阶提出了多重挑战，包括解流形的Kolmogorov $N$-宽度衰减缓慢、激波形成以及解对参数的光滑依赖性丧失。这些问题限制了传统基于本征正交分解的降阶模型的性能。我们基于低秩张量分解的tROM方法，从高保真快照构建参数到解的映射，并通过局部POD过程构建局部化降阶基。我们将该方法应用于干床和湿床两种问题设置，结果表明：tROM的非插值变体结合临界参数值附近的切比雪夫采样，能有效捕捉参数依赖行为，且显著优于标准POD-ROM。这在湿床工况中尤为明显——传统POD-ROM对激波的分辨率较差且会出现伪振荡现象。