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.

💡 Research Summary

This paper presents the development and application of a parametric tensor Reduced-Order Model (tROM) for simulating the shallow-water dam-break problem, a canonical hyperbolic system notorious for its challenges in model reduction.

The core challenge lies in the nature of hyperbolic PDEs like the shallow-water equations. Their solution manifolds often exhibit a slow decay of the Kolmogorov N-width, meaning they cannot be efficiently approximated by low-dimensional linear subspaces. Furthermore, solutions develop shocks and display a loss of smooth dependence on parameters—specifically, the derivative of the solution with respect to the downstream water depth (h_R) becomes unbounded as (h_R \to 0). These issues severely limit the performance of traditional projection-based ROMs that rely on global Proper Orthogonal Decomposition (POD), leading to poor shock resolution and spurious oscillations.

To overcome these limitations, the authors propose a tensor-based approach. The method organizes high-fidelity snapshot data—solutions for different parameter values ((h_L, h_R)) and time instances—into a single 4D tensor (space × parameter1 × parameter2 × time). A low-rank Tucker decomposition is then applied to this tensor, extracting a core tensor and factor matrices. This decomposition implicitly defines a “universal” subspace containing information from all training snapshots.

For a new, unseen parameter value, the tROM constructs a local reduced basis tailored to that specific point in parameter space. The paper investigates two variants: 1) Interpolatory tROM, which uses tensor interpolation to generate the local basis, and 2) Non-interpolatory tROM, which instead concatenates data from the four nearest parameter samples and performs a local POD. The non-interpolatory method proves superior, as it avoids assumptions of smooth interpolation across parameter space, which is invalidated by the solution’s irregular dependence on (h_R). An effective sampling strategy using Chebyshev nodes near critical parameter values (small (h_R)) is also emphasized to better capture rapid solution changes.

The method is tested extensively on 1D dam-break problems, both for dry-bed ((h_R=0)) and wet-bed ((h_R>0)) conditions. Results demonstrate that the non-interpolatory tROM significantly outperforms both standard POD-ROM and interpolatory tROM. While POD-ROM fails to accurately resolve shock waves, showing phase errors and large oscillations, the tROM captures shock location and strength with high fidelity. The advantage is most pronounced in the challenging wet-bed case.

Finally, the framework is successfully extended to a more complex 2D wet-bed dam-break problem with five parameters, including topography and bottom friction. The tROM maintains its accuracy advantage over POD-ROM in this higher-dimensional setting, demonstrating the method’s robustness and scalability.

In conclusion, this work provides a compelling case for using low-rank tensor decompositions and local basis construction to build accurate ROMs for hyperbolic systems with parameter-dependent discontinuities. The non-interpolatory tROM offers a mathematically grounded and practically effective strategy to tackle the slow N-width decay and parameter irregularities that plague traditional linear reduction techniques.