The influence of different boundary conditions on the spectral properties of the incompressible Navier-Stokes equations is investigated. By using the Fourier-Laplace transform technique, we determine the spectra, extract the decay rate in time, and investigate the dispersion relation. In contrast to an infinite domain, where only diffusion affects the convergence, we show that also the propagation speed influence the rate of convergence to steady state for a bounded domain. Once the continuous equations are analyzed, we discretize using high-order finite-difference operators on summation-by-parts form and demonstrate that the continuous analysis carries over to the discrete setting. The theoretical results are verified by numerical experiments, where we highlight the necessity of high accuracy for a correct description of time-dependent phenomena.

研究了不同边界条件对不可压缩Navier-Stokes方程谱特性的影响。通过使用傅里叶-拉普拉斯变换技术，我们确定光谱，及时提取衰减率，并研究色散关系。与仅扩散影响收敛的无限域相反，我们表明，对于有界域，传播速度也会影响收敛到稳态的速率。一旦分析了连续方程，我们就可以使用逐部分求和形式的高阶有限差分算子进行离散化，并证明连续分析可以推广到离散设置。理论结果通过数值实验得到验证，其中我们强调了正确描述随时间变化现象的高精度的必要性。