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Implicit Boundary Conditions in Partial Differential Equations Discretizations: Identifying Spurious Modes and Model Reduction

Pascal R Karam, Bassam Bamieh

TL;DR

This work reframes spurious eigenvalues in discretized PDEs as a consequence of neglecting implicit boundary conditions that accompany homogeneous BCs, and develops a unifying DAE perspective. It introduces two intrinsic quality tests—the derivative-violation measure and the Grassmann distance between subspaces—to identify and quantify the accuracy of computed eigenpairs without prior spectrum knowledge, and extends these ideas to generalized eigenvalue problems. By enforcing implicit constraints through a controllable truncation of constraints and filtering modes with the proposed quality metrics, the authors demonstrate both elimination of spurious modes in select cases and robust model reduction by discarding low-quality modes. The approach provides a general, practical framework for diagnosing discretization quality and constructing reduced-order models for PDE-based dynamics, with demonstrated applicability to canonical problems including Orr-Sommerfeld-type systems and 1D acoustic waves.

Abstract

We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.

Implicit Boundary Conditions in Partial Differential Equations Discretizations: Identifying Spurious Modes and Model Reduction

TL;DR

This work reframes spurious eigenvalues in discretized PDEs as a consequence of neglecting implicit boundary conditions that accompany homogeneous BCs, and develops a unifying DAE perspective. It introduces two intrinsic quality tests—the derivative-violation measure and the Grassmann distance between subspaces—to identify and quantify the accuracy of computed eigenpairs without prior spectrum knowledge, and extends these ideas to generalized eigenvalue problems. By enforcing implicit constraints through a controllable truncation of constraints and filtering modes with the proposed quality metrics, the authors demonstrate both elimination of spurious modes in select cases and robust model reduction by discarding low-quality modes. The approach provides a general, practical framework for diagnosing discretization quality and constructing reduced-order models for PDE-based dynamics, with demonstrated applicability to canonical problems including Orr-Sommerfeld-type systems and 1D acoustic waves.

Abstract

We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.
Paper Structure (13 sections, 2 theorems, 50 equations, 6 figures, 2 algorithms)

This paper contains 13 sections, 2 theorems, 50 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation: Implicit Boundary Conditions and their Approximations
  3. Finite-Dimensional Setting: Differential Algebraic Equations (DAEs)
  4. Implicit/Hidden Boundary Conditions in PDEs
  5. Eliminating and Identifying Spurious Modes
  6. Enforcing Implicit Constraints
  7. Tests for Identifying Spurious Modes
  8. Time Derivatives of Eigenfunctions at Boundaries as a Violation Measure
  9. Angle Criterion for Eigenvalue Problems
  10. Generalized Eigenvalue Problems
  11. Combining Quality Criteria with Implicit Constraints
  12. Model Reduction by Elimination of Low Quality Modes
  13. Proof of Lemma \ref{['DAE2.lemma']}

Key Result

Lemma 1

The DAE (zd_A.eq)-(Cz_zero.eq) has a solution iff the initial condition is in the null space ${\sf Nu}\!\left( \mathcal{O}_n \right)$ of the matrix where $n$ is the dimension of $z$. When that condition holds, the solution remains in that subspace for all $t\geq 0$. ${\sf Nu}\!\left( \mathcal{O}_n \right)$ is the largest $A$-invariant subspace contained in ${\sf Nu}\!\left( C \right)$.

Figures (6)

  • Figure 1: Explicitly enforcing implicit constraints through $\mathcal{O}_k$ can remove traditionally spurious modes (eigenvalues with erroneously large real part) in spectral collocation schemes. When discretizing (\ref{['canuto_sys.eq']}) using a Chebyshev collocation scheme using differentiation matrices of size: $N = 16$ (\ref{['canuto_k_n16.fig']}), $N = 32$ (\ref{['canuto_k_n32.fig']}) and $N = 64$ (\ref{['canuto_k_n64.fig']}) there always exists a sufficiently large $k$ such that the traditionally spurious modes disappear. This is indicated by the sharp drop in real component of the maximum eigenvalue error. Here $m$ corresponds to the index of the maximum absolute eigenvalue error $m := \rm{argmax}_i\{ | \tilde{\lambda}_i - \bar{\lambda}_i | \}$ However the growth in minimum absolute eigenvalue error for increasing $k$ indicates that applying a large number of implicit constraints contaminates the remaining spectrum. These competing behaviors make it difficult to apply this method in general.
  • Figure 2: Analysis of the "quality" of the spectrum of system (\ref{['canuto_disc.eq']}) discretized using a Chebyshev collocation scheme with $N = 16$ (\ref{['Canuto_1D_rel_error16.fig']}), $N = 64$ (\ref{['Canuto_1D_rel_error64.fig']}), and $N = 128$ (\ref{['Canuto_1D_rel_error128.fig']}) nodes per field. For each discretization size, the three error criteria are plotted versus the imaginary component of the respective eigenvalue. The relative error $(\bar{\lambda}-\tilde{\lambda})/\bar{\lambda}$, the first implicit constraint violation measure $\|s_l\|$ (derivative test), and the Grassmann distance $\theta_l$ are shown for all three cases. Boundary conditions are implemented using the automated procedure (Step 1) of Algorithm \ref{['Alg_second']}. In all three cases, both error criteria the relative eigenvalue error well. Note that because the Grassmann distance is root sum square of two angles, a Grassmann distance of $1e \text{-} 6$ equates to an error of order machine precision.
  • Figure 3: Spectrum of Orr-Sommerfeld operator for Poiseuille flow with R = 10, 000, $\alpha$ = 1. The system is discretized using a Chebyshev collocation scheme with a grid size of N = 130. The approximate system contains a large number of spurious modes which are clearly identified by the Grassmann distance criterion $\theta$. Inset shows the classic Orr-Sommerfeld spectrum which is captured well by the discretized problem.
  • Figure 4: Approximate eigenvalue $\tilde{\lambda}$ corresponding to the Tollmien-Schlichting (T-S) wave of the Orr-Sommerfeld operator for Poiseuille flow discretized using Chebyshev collocation with R = 10,000, $\alpha = 1$, grid sizes N = $50,\ldots, 150$. This is the same configuration as in Orzag1971 who reported the eigenvalue corresponding to the T-S wave to be $0.00373967 - 0.23752649i$. Our reported eigenvalue with smallest Grassmann distance $\theta$ corresponds best with that in Orzag1971.
  • Figure 5: Contours of the true eigenvalue error (Column (\ref{['Canuto_1D_contour_eigerror.fig']})), boundary derivative (Column (\ref{['Canuto_1D_contour_xdot.fig']})) and Grassmann distance (Column (\ref{['Canuto_1D_contour_grassmann.fig']})) as in system (\ref{['canuto_disc.eq']}) discretized using a Chebyshev collocation scheme with $N = 16$ (first row), $N = 64$ (second row) and $N = 128$ (third row). For each grid size, level sets of the error criteria are shown for various truncation orders of the $\mathcal{O}^{k}$. This demonstrates that for small $k$, both $\left\lVert s_l \right\rVert$ and $\theta_l$ both provide good approximations to the eigenvalue error. However, $\left\lVert s_l \right\rVert$ degrades at higher $k$ (due to the ill-conditioned derivative matrices) and no longer provides insight into the mode quality. Furthermore, increasing $k$ does not improve the subset of the spectrum that is well captured as shown by the nearly vertical contours in both (\ref{['Canuto_1D_contour_eigerror.fig']}) and (\ref{['Canuto_1D_contour_grassmann.fig']}) as $k$ is increased.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2: Constraint-free Transformation of the DAE