A parametric tensor ROM for the shallow water dam break problem

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 1D dry-bed and wet-bed problems and 2D wet-bed problem with topography and bottom friction, 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.