We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem of Fuchsian type by a sequence of regular Cauchy problems, which we next discretize by pseudo-spectral and Runge-Kutta techniques. Our main contribution is a detailed analysis of the numerical error which has two distinct sources, and our main proposal here is to keep in balance the errors arising at the continuum and at the discrete levels of approximation. We present numerical experiments which strongly support our theoretical conclusions. This strategy is finally applied to applied to compressible fluid flows evolving on a Kasner spacetime, and we numerically demonstrate the nonlinear stability of such flows, at least in the so-called sub-critical regime identified earlier by the authors.

我们考虑了一类Fuchsian方程，例如，它描述了宇宙时空中可压缩流体流动的演变。使用线的方法，当数据施加于宇宙奇异点并且从奇异超曲面执行演化时，我们针对奇异初值问题引入了一种数值算法。我们通过一系列常规柯西问题来近似Fuchsian型奇异柯西问题，然后我们通过伪光谱和龙格-库塔技术将其离散化。我们的主要贡献是对数值误差的详细分析，它有两个不同的来源，而我们的主要建议是保持平衡在连续和近似离散水平上出现的误差。我们提出的数值实验强烈支持我们的理论结论。该策略最终应用于应用于随Kasner时空演化的可压缩流体流，并且我们至少在作者先前确定的所谓亚临界状态下，通过数值方法证明了这种流的非线性稳定性。