Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners

TL;DR

The paper addresses reliable spectral computation for convective spiral waves, where the resolvent norm can blow up with domain size. It introduces exponential weights as inexpensive preconditioners, enabling uniform resolvent bounds and accurate eigenvalue extraction by relating the optimal weight $\eta(\lambda)$ to a simpler one-dimensional far-field problem via spatial eigenvalues $\nu_j(\lambda)$. The authors provide a rigorous spatial-dynamics framework to prove uniform resolvent estimates and the exponential growth/decay dichotomies, and they translate these results into a practical numerical algorithm tested on the Barkley model, showing significantly improved accuracy and conditioning. This work offers a principled, scalable method for stable spectral analysis of spiral waves in extended domains, with potential applicability to other convection-dominated spectra problems.

Abstract

The stability of nonlinear waves on spatially extended domains is commonly probed by computing the spectrum of the linearization of the underlying PDE about the wave profile. It is known that convective transport, whether driven by the nonlinear pattern itself or an underlying fluid flow, can cause exponential growth of the resolvent of the linearization as a function of the domain length. In particular, sparse eigenvalue algorithms may result in inaccurate and spurious spectra in the convective regime. In this work, we focus on spiral waves, which arise in many natural processes and which exhibit convective transport. We prove that exponential weights can serve as effective, inexpensive preconditioners that result in resolvents that are uniformly bounded in the domain size and that stabilize numerical spectral computations. We also show that the optimal exponential rates can be computed reliably from a simpler asymptotic problem posed in one space dimension.

Figures (6)

• Figure 1: Shown are the eigenvalues of $\mathcal{L}_{R,\eta}$ with $c=1$ for different values of $R$ with $\eta=0,0.25,0.5$. Eigenvalues not visible lie in $\Sigma_\mathrm{abs}=(-\infty,-c^2/4]$. The numerical spectrum for $R=800$ with $\eta=0.5=c/2$ agrees with the theoretical spectrum within $5\times10^{-3}$ accuracy.
• Figure 1: (a) Inaccurate computation of point spectra of $\mathcal{L}_R$. Eigenvalues of spiral waves show divergence from $\Sigma_{\text{abs}}$ for increasing $R$ rather than the expected convergence. The spiral profiles capture the $u$-component of the Barkley model. (b)&(c) Eigenvalues approach the anticipated limit points upon appropriate selection of the exponential weight.
• Figure 2: Shown are the pseudospectrum $\Lambda_\epsilon(\mathcal{L}_R)$, the Fredholm boundary $\Sigma_{\mathrm{FB},\eta}$, and the numerical eigenvalues for a range of weight values $\eta$ with $c=1$. The color scale reflects the minimal singular value of the finite-difference matrix for $\mathcal{L}_R-\lambda$ on a $\log_{10}$ scale and therefore provides the pseudospectrum contours of $\Lambda_\epsilon$. Eigenvalues were found using MATLAB's direct solver eig.
• Figure 2: Comparison of $\epsilon$-pseudospectra and eigenvalues of the operator $\mathcal{L}_R$ (top row) and $\mathcal{L}_R^{\eta}$ (bottom row) for $\eta = -1.5$. The three columns correspond, from left to right, to disks of radius $R = 25, 50, 75$. Red curves show $\Sigma_\mathrm{abs}$.
• Figure 3: Leading spatial eigenvalues $\nu_j(\lambda)$ shown in (b) as $\lambda$ moves along the path indicated by the horizontal dashed arrow in (a). Green and red markers in (b) indicate $\nu_j(\lambda)$ for $\lambda = 0.1 + 0.5 \mathrm{i}$ (green) and $\lambda = -1 + 0.5 \mathrm{i}$ (red). Spatial eigenvalues $\nu_{-1}(\lambda)$ and $\nu_0(\lambda)$ relevant for $J_0(\lambda)$ are labeled.

Theorems & Definitions (21)

• Proposition 2.1: trefethen2, ss-trunc
• Proposition 2.2: ss-trunc
• Definition 3.1: Admissible wave trains
• Definition 3.2: Absolute spectrum
• Definition 3.3: Spiral waves
• Definition 3.4: Extended point spectrum of spiral waves
• Definition 3.5: Transverse spiral waves
• Definition 3.6: Non-degenerate boundary sinks
• Definition 3.7: Extended point spectrum of boundary sinks
• Theorem 3.8