Efficient numerical computation of spiral spectra with exponentially-weighted preconditioners

TL;DR

This paper introduces exponential-weighted preconditioners to stabilize and improve the accuracy of spectral computations for spiral waves in PDEs, addressing issues caused by convective transport and domain size effects.

Contribution

It demonstrates that exponential weights serve as effective preconditioners for spectral stability analysis of spiral waves, enabling reliable computations in large domains.

Findings

Exponential weights stabilize resolvent bounds in large domains.

Optimal exponential rates can be derived from a simplified 1D asymptotic problem.

Preconditioners improve accuracy of eigenvalue computations in convective regimes.

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.