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A parametric tensor ROM for the shallow water dam break problem

Md Rezwan Bin Mizan, Maxim Olshanskii, Ilya Timofeyev

TL;DR

The paper advances tensor reduced-order modeling (tROM) for parametric shallow-water dam-break problems by employing a Tucker-based low-rank decomposition to build parameter-adaptive local bases. It contrasts interpolatory and non-interpolatory tROM variants against POD-ROM, showing that non-interpolatory tROM with Chebyshev sampling near critical parameter values robustly captures shock-dominated dynamics and remains effective as the wet-bed depth $h_R$ approaches zero. The framework is demonstrated in 1D (dry- and wet-bed) and 2D SWE settings, highlighting improved shock resolution and reduced spurious oscillations, especially in the wet-bed regime. The results highlight the potential of tROM to address hyperbolic, parameter-dependent PDEs with slow $N$-width decay and complex solution regularity, while outlining directions for further local-time localization and tensor-completion enhancements.

Abstract

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.

A parametric tensor ROM for the shallow water dam break problem

TL;DR

The paper advances tensor reduced-order modeling (tROM) for parametric shallow-water dam-break problems by employing a Tucker-based low-rank decomposition to build parameter-adaptive local bases. It contrasts interpolatory and non-interpolatory tROM variants against POD-ROM, showing that non-interpolatory tROM with Chebyshev sampling near critical parameter values robustly captures shock-dominated dynamics and remains effective as the wet-bed depth approaches zero. The framework is demonstrated in 1D (dry- and wet-bed) and 2D SWE settings, highlighting improved shock resolution and reduced spurious oscillations, especially in the wet-bed regime. The results highlight the potential of tROM to address hyperbolic, parameter-dependent PDEs with slow -width decay and complex solution regularity, while outlining directions for further local-time localization and tensor-completion enhancements.

Abstract

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 -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.

Paper Structure

This paper contains 29 sections, 49 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Methods and Setup
  3. 1D Shallow Water Equations
  4. Parametrized initial conditions.
  5. Boundary conditions.
  6. Parameters sampling.
  7. 1D SWE Reduced-Order Modeling
  8. 1D SWE Tensor ROM
  9. LRTD
  10. 1D SWE Interpolatory tROM
  11. 1D SWE Non-Interpolatory tROM
  12. 1D SWE POD-ROM
  13. Implementation
  14. Offline Stage
  15. Online Stage
  16. ...and 14 more sections

Figures (14)

  • Figure 1: Water depth profiles evolution in time for the cases of dry and wet-bed. Snapshots are shown every $\Delta t = 0.5$. Click the plots for full animation.
  • Figure 2: Performance metrics for varying the local threshold $\epsilon_{\text{loc}}$ (from $4.0 \times 10^{-2}$ to $1.0 \times 10^{-3}$) in the non-interpolatory tROM ($h_{\!L} = 27 \, \text{m}$): (a) water depth profiles at final time for $h_{\!R} = 0.14 \, \text{m}$, with a zoom-in of the step region showing improved shock front resolution as $\epsilon_{\text{loc}}$ decreases; (b) water depth profiles at final time for $h_{\!R} = 3.00 \, \text{m}$, with a zoom-in of the step region indicating smoother transitions but similar trends; (c) relative errors in $h$ versus the local threshold with $h_{\!R} = 0.14 \, \text{m}$; (d) relative errors in $h$ versus the local threshold with $h_{\!R} = 3.00 \, \text{m}$, exhibiting a similar error reduction trend; (e) ROM dimensions ($\ell_h, \ell_{q}$) for $h_{\!R} = 0.14 \, \text{m}$, increasing with smaller $\epsilon_{\text{loc}}$ to retain more basis vectors; (f) ROM dimensions ($\ell_h, \ell_{q}$) for $h_{\!R} = 3.00 \, \text{m}$, also increasing with smaller $\epsilon_{\text{loc}}$.
  • Figure 3: Comparison of water depth profiles and relative $L^2$ errors for $h_{\!L} = 25\,\text{m}$ using Chebyshev and uniform node distributions ($N_L = 13$, $N_R = 9$) in the non-interpolatory tROM: (a) final-time depth profiles for $h_{\!R} = 0.05\,\text{m}$; (b) final-time depth profiles for $h_{\!R} = 0.3\,\text{m}$; (c) relative $L^2(0,T;L^2(\Omega))$ error over time for $h_{\!R} = 0.05\,\text{m}$; (d) same error metric for $h_{\!R} = 0.3\,\text{m}$.
  • Figure 4: Water depth profiles and error evolution over time for specific parameter pairs ($N_L = 13, N_R = 9, h_{\!L} = 25 \, \text{m}$), comparing interpolatory and non-interpolatory tROMs : (a) water depth profiles at final time for $h_{\!R} = 0.05 \, \text{m}$; (b) water depth profiles at final time for $h_{\!R} = 2 \, \text{m}$; (c) relative error evolution in over time for $h_{\!R} = 0.05 \, \text{m}$; (d) relative error evolution over time for $h_{\!R} = 2.00 \, \text{m}$.
  • Figure 5: Water depth profiles at final time for different parameter pairs, comparing non-interpolatory tROM and POD ROM (with equal ROM dimensions derived from tROM using $\epsilon_{\text{loc}} = 4.0 \times 10^{-3}$) with Chebyshev nodes: (a) for $h_{\!L} = 12, h_{\!R} = 0$; (b) for $h_{\!L} = 12, h_{\!R} = 7$; (c) for $h_{\!L} = 15, h_{\!R} = 4$; (d) for $h_{\!L} = 18, h_{\!R} = 0$; (e) for $h_{\!L} = 26, h_{\!R} = 0.14$; (f) for $h_{\!L} = 26, h_{\!R} = 7$.
  • ...and 9 more figures