The susceptibility of timestepping algorithms to numerical instabilities is an important consideration when simulating partial differential equations (PDEs). Here we identify and analyze a pernicious numerical instability arising in pseudospectral simulations of nonlinear wave propagation resulting in finite-time blow-up. The blow-up time scale is independent of the spatial resolution and spectral basis but sensitive to the timestepping scheme and the timestep size. The instability appears in multi-step and multi-stage implicit-explicit (IMEX) timestepping schemes of different orders of accuracy and has been found to manifest in simulations of soliton solutions of the Korteweg-de Vries (KdV) equation and traveling wave solutions of a nonlinear generalized Klein-Gordon equation. Focusing on the case of KdV solitons, we show that modal predictions from linear stability theory are unable to explain the instability because the spurious growth from linear dispersion is small and nonlinear sources of error growth converge too slowly in the limit of small timestep size. We then develop a novel multi-scale asymptotic framework that captures the slow, nonlinear accumulation of timestepping errors. The framework allows the solution to vary with respect to multiple time scales related to the timestep size and thus recovers the instability as a function of a slow time scale dictated by the order of accuracy of the timestepping scheme. We show that this approach correctly describes our simulations of solitons by making accurate predictions of the blow-up time scale and transient features of the instability. Our work demonstrates that studies of long-time simulations of nonlinear waves should exercise caution when validating their timestepping schemes.

翻译：时间步进算法对数值不稳定性的敏感性是模拟偏微分方程时的重要考量因素。本文识别并分析了一种在非线性波传播的伪谱模拟中出现的、导致有限时间爆发的有害数值不稳定性。爆发时间尺度与空间分辨率和谱基无关，但对时间步进方案和时间步长敏感。该不稳定性出现在不同精度阶数的多步和多级隐式-显式时间步进方案中，并已在Korteweg-de Vries方程的孤子解以及非线性广义Klein-Gordon方程的行波解模拟中被发现。聚焦于KdV孤子情形，我们证明线性稳定性理论的模态预测无法解释该不稳定性，因为线性色散引起的寄生增长较小，且非线性误差增长源在小时间步长极限下收敛过慢。随后我们建立了一个新颖的多尺度渐近框架，该框架能够捕捉时间步进误差的缓慢非线性累积。该框架允许解相对于与时间步长相关的多个时间尺度变化，从而将不稳定性恢复为受时间步进方案精度阶数支配的慢时间尺度函数。我们通过精确预测爆发时间尺度及不稳定性的瞬态特征，证明该方法能正确描述我们的孤子模拟。本研究表明，在验证非线性波长时间模拟的时间步进方案时需保持谨慎。