论文标题
用于宇宙学空间上的紫红色方程和流体流的数值算法
A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes
论文作者
论文摘要
我们考虑了一类紫红色方程，例如描述了可压缩流体在宇宙学时空上的演变。使用线条方法，当数据对宇宙学奇异性施加并从奇异性超表面进行进化时，我们引入了一种数值算法，以实现奇异初始值问题。我们通过一系列常规的考奇问题近似五叶族类型的奇异凯奇问题，然后我们接下来通过伪谱和runge-kutta技术离散。我们的主要贡献是对具有两个不同来源的数值误差的详细分析，我们的主要建议是保持平衡在连续体和离散近似级别处产生的误差。我们提出的数值实验强烈支持我们的理论结论。最终将该策略应用于在Kasner时空上演变的可压缩流体流，我们从数值上证明了此类流的非线性稳定性，至少在作者早期所谓的所谓的亚临界方案中。
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.
