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First-order empirical interpolation method for real-time solution of parametric time-dependent nonlinear PDEs

Ngoc Cuong Nguyen

Abstract

We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate reduced-order models. Proper orthogonal decomposition is used to construct a reduced-basis (RB) space which provides a rapidly convergent approximation of the parametric solution manifold. The Galerkin projection is employed to reduce the dimensionality of the problem by projecting the weak formulation of the governing PDEs onto the RB space. A major challenge in model reduction for nonlinear PDEs is the efficient treatment of nonlinear terms, which we address by unifying the implementation of several hyperreduction methods. We introduce a first-order empirical interpolation method to approximate the nonlinear terms and recover the computational efficiency. We demonstrate the effectiveness of our methodology through its application to the Allen-Cahn equation, which models phase separation processes, and the Buckley-Leverett equation, which describes two-phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed approach.

First-order empirical interpolation method for real-time solution of parametric time-dependent nonlinear PDEs

Abstract

We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate reduced-order models. Proper orthogonal decomposition is used to construct a reduced-basis (RB) space which provides a rapidly convergent approximation of the parametric solution manifold. The Galerkin projection is employed to reduce the dimensionality of the problem by projecting the weak formulation of the governing PDEs onto the RB space. A major challenge in model reduction for nonlinear PDEs is the efficient treatment of nonlinear terms, which we address by unifying the implementation of several hyperreduction methods. We introduce a first-order empirical interpolation method to approximate the nonlinear terms and recover the computational efficiency. We demonstrate the effectiveness of our methodology through its application to the Allen-Cahn equation, which models phase separation processes, and the Buckley-Leverett equation, which describes two-phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed approach.
Paper Structure (16 sections, 1 theorem, 51 equations, 5 figures, 4 tables, 2 algorithms)

This paper contains 16 sections, 1 theorem, 51 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model reduction methods
  3. Finite element approximation
  4. Proper orthogonal decomposition
  5. Galerkin-Newton method
  6. Hyperreduced Galerkin-Newton method
  7. First-Order Empirical Interpolation Method
  8. Empirical interpolation procedure
  9. Basis functions
  10. Interpolation points
  11. Error estimate of the first-order empirical interpolation method
  12. A simple test case
  13. Numerical Experiments
  14. Buckley-Leverett equation
  15. Allen-Cahn equation
  16. ...and 1 more sections

Key Result

Theorem 1

If $g(u_N(\bm x, t, \bm \mu)) \in \Psi_{M+P}$ for $P \in \mathbb{N}_{+}$, then the interpolation error $\varepsilon_M(t,\bm \mu) \equiv \|g(u_N(\bm x, t, \bm \mu)) - g_M(\bm x, t, \bm \mu) \|_{L^\infty(\Omega)}$ is bounded by where $e_j(t,\bm \mu), 1 \le j \le P,$ solve the following linear system

Figures (5)

  • Figure 1: Plots of $g(u(x,t,\mu))$ as a function of $x$ and $t$ for $\mu = 0$ and $\mu =10$.
  • Figure 2: Mean absolute error as a function of $M$ and $L$ for $J = 6$ and $J =12$.
  • Figure 3: Comparison of accuracy between the GN method and the FOEIM-GN method with three different values of $L$ and $M$.
  • Figure 4: Comparison of accuracy between the GN method and the FOEIM-GN method with four different values of $L$ and $M$.
  • Figure 5: Evolution of the FOEIM-GN solution with $N = 50, M = 100$, and $L=1$ (top row) or $L = 3$ (bottom row) for $\mu = 0.34$.

Theorems & Definitions (1)

  • Theorem 1