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Well-balanced POD-based reduced-order models for finite volume approximation of hyperbolic balance laws

I. Gómez-Bueno, E. D. Fernández-Nieto, S. Rubino

TL;DR

The paper develops POD-based reduced-order models for 1D hyperbolic balance laws discretized by finite volumes, combining DEIM with Proper Interval Decomposition to efficiently handle nonlinearities. A key theoretical contribution is a well-balanced ROM result: if the full-order model is exactly well-balanced for a stationary state, the ROM inherits this property, with the snapshot structure sometimes collapsing to a single POD mode. Numerically, the authors show that PID-DEIM ROMs can achieve substantial speed-ups while maintaining accuracy across transport, Burgers, and shallow-water systems, including parameter-dependent scenarios with Manning friction. The work demonstrates improved performance over time-averaging approaches, especially near discontinuities, and provides a framework for predictive ROMs in parameterized hyperbolic problems with potential applications to geophysical flows.

Abstract

This paper introduces a reduced-order modeling approach based on finite volume methods for hyperbolic systems, combining Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and Proper Interval Decomposition (PID). Applied to systems such as the transport equation with source term, non-homogeneous Burgers equation, and shallow water equations with non-flat bathymetry and Manning friction, this method achieves significant improvements in computational efficiency and accuracy compared to previous time-averaging techniques. A theoretical result justifying the use of well-balanced Full-Order Models (FOMs) is presented. Numerical experiments validate the approach, demonstrating its accuracy and efficiency. Furthermore, the question of prediction of solutions for systems that depend on some physical parameters is also addressed, and a sensitivity analysis on POD parameters confirms the model's robustness and efficiency in this case.

Well-balanced POD-based reduced-order models for finite volume approximation of hyperbolic balance laws

TL;DR

The paper develops POD-based reduced-order models for 1D hyperbolic balance laws discretized by finite volumes, combining DEIM with Proper Interval Decomposition to efficiently handle nonlinearities. A key theoretical contribution is a well-balanced ROM result: if the full-order model is exactly well-balanced for a stationary state, the ROM inherits this property, with the snapshot structure sometimes collapsing to a single POD mode. Numerically, the authors show that PID-DEIM ROMs can achieve substantial speed-ups while maintaining accuracy across transport, Burgers, and shallow-water systems, including parameter-dependent scenarios with Manning friction. The work demonstrates improved performance over time-averaging approaches, especially near discontinuities, and provides a framework for predictive ROMs in parameterized hyperbolic problems with potential applications to geophysical flows.

Abstract

This paper introduces a reduced-order modeling approach based on finite volume methods for hyperbolic systems, combining Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and Proper Interval Decomposition (PID). Applied to systems such as the transport equation with source term, non-homogeneous Burgers equation, and shallow water equations with non-flat bathymetry and Manning friction, this method achieves significant improvements in computational efficiency and accuracy compared to previous time-averaging techniques. A theoretical result justifying the use of well-balanced Full-Order Models (FOMs) is presented. Numerical experiments validate the approach, demonstrating its accuracy and efficiency. Furthermore, the question of prediction of solutions for systems that depend on some physical parameters is also addressed, and a sensitivity analysis on POD parameters confirms the model's robustness and efficiency in this case.
Paper Structure (27 sections, 2 theorems, 80 equations, 16 figures, 6 tables)

This paper contains 27 sections, 2 theorems, 80 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Hyperbolic PDE systems: full-order models
  3. Full-order models: examples
  4. Transport equation with a linear source term
  5. Burgers equation with a nonlinear source term
  6. Shallow water system with non-flat bathymetry and Manning friction
  7. POD-based reduced-order models for hyperbolic PDE systems
  8. Dealing with nonlinearities
  9. Reduced-order models: some examples
  10. Transport equation with a linear source term
  11. Burgers equation with a nonlinear source term
  12. Shallow water system with non-flat bathymetry Manning friction
  13. Modified Lax-Friedrichs ROM
  14. HLL ROM
  15. Well-balanced property
  16. ...and 12 more sections

Key Result

Theorem 1

Let us suppose that the numerical scheme met_num_FOM_operador for the FOM is well-balanced for a stationary solution $W^*$, i.e., $W_i^n=W_i^1, \, \forall i, \, \forall n$, where $W_i^1$ are the cell-averages of $W^*$. Then, the associated ROM is also well-balanced for $W^*$.

Figures (16)

  • Figure 1: Well-balanced property check: transport equation. Initial condition (left) and differences between the initial and final numerical solutions for a 200-cell mesh at $T_f=10$ with the ROM (right).
  • Figure 2: Well-balanced property check: Burgers equation. Initial condition (left) and differences between the initial and final numerical solutions for a 200-cell mesh at $T_f=10$ with the ROM (right).
  • Figure 3: Well-balanced property check: shallow water equations. Initial condition: free surface $\eta$ and bottom (left) and velocity $u$ (right).
  • Figure 4: Well-balanced property check: shallow water equations. Differences between the initial and final numerical solutions for a 200-cell mesh at $T_f=10$ with the ROM: free surface $\eta$ (left) and velocity $u$ (right).
  • Figure 5: Transport equation: small perturbation of a stationary solution. Initial condition
  • ...and 11 more figures

Theorems & Definitions (5)

  • Definition 1
  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1