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.

翻译：在空间扩展域上非线性波稳定性的研究通常通过计算波剖面处偏微分方程线性化算子的谱来进行。已知对流输运（无论由非线性模式本身驱动还是底层流体流动驱动）会导致线性化解算子的指数增长，且该增长与域长度相关。具体而言，在对流主导区域中，稀疏特征值算法可能产生不准确且虚假的谱。本研究聚焦于螺旋波——这类波出现在众多自然过程中且表现出对流输运特性。我们证明指数权重可作为有效且低成本的预条件子，使解算子在域尺寸上一致有界，从而稳定数值谱计算。同时证明最优指数速率可通过一维空间中更简化的渐近问题可靠计算得出。