二阶双曲型Fuchsian方程组Gowdy时空与Fuchsian数值算法
Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm
摘要：
这是一系列致力于奇异初值问题的二阶双曲型Fuchsian系统的第二部分。在第一部分中，我们定义并研究了这类系统，并在加权Sobolev空间中建立了一个适定性理论。这个理论在这里适用于真空爱因斯坦方程的高迪时空承认，根据定义，两个Killing场满足一定的几何条件。我们恢复，更直接和更简单的参数，适定性的结果建立了较早的Rendall和合作者。此外，在本文中，我们引入了一个自然的近似方案，我们称之为Fuchsian数值算法，并直接激发我们的一般理论。该算法提供了奇异初值问题解的高精度数值逼近。特别是，Gowdy时空类考虑中，各种数值实验表明所提出的方法的兴趣和效率。最后，作为一个应用，我们数值构造了一个光滑的，不完整的，非紧的柯西视界的Gowdy时空。
英文摘要：
This is the second part of a series devoted to the singular initial value problem for second-order hyperbolic Fuchsian systems. In the first part, we defined and investigated this general class of systems, and we established a well-posedness theory in weighted Sobolev spaces. This theory is applied here to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition, two Killing fields satisfying certain geometric conditions. We recover, by more direct and simpler arguments, the well-posedness results established earlier by Rendall and collaborators. In addition, in this paper we introduce a natural approximation scheme, which we refer to as the Fuchsian numerical algorithm and is directly motivated by our general theory. This algorithm provides highly accurate, numerical approximations of the solution to the singular initial value problem. In particular, for the class of Gowdy spacetimes under consideration, various numerical experiments are presented which show the interest and efficiency of the proposed method. Finally, as an application, we numerically construct Gowdy spacetimes containing a smooth, incomplete, non-compact Cauchy horizon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
