This paper presents a convergence analysis of a Krylov subspace spectral (KSS) method applied to a 1-D wave equation in an inhomogeneous medium. It will be shown that for sufficiently regular initial data, this KSS method yields unconditional stability, spectral accuracy in space, and second-order accuracy in time, in the case of constant wave speed and a bandlimited reaction term coefficient. Numerical experiments that corroborate the established theory are included, along with an investigation of generalizations, such as to higher space dimensions and nonlinear PDEs, that features performance comparisons with other Krylov subspace-based time-stepping methods. This paper also includes the first stability analysis of a KSS method that does not assume a bandlimited reaction term coefficient.

翻译：本文针对一维非均匀介质中的波动方程，提出了一种Krylov子空间谱（KSS）方法的收敛性分析。研究表明，在波速恒定且反应项系数带限的条件下，对于足够光滑的初始数据，该KSS方法具有无条件稳定性、空间谱精度及时间二阶精度。文中通过数值实验验证了上述理论，并探讨了该方法在高维空间及非线性偏微分方程中的推广，同时与其他基于Krylov子空间的时间步进方法进行了性能比较。此外，本文首次给出了不假设反应项系数带限的KSS方法的稳定性分析。