Solving the Einstein Constraints Numerically on Compact Three-Manifolds Using Hyperbolic Relaxation
arxiv(2024)
摘要
The effectiveness of the hyperbolic relaxation method for solving the
Einstein constraint equations numerically is studied here on a variety of
compact orientable three-manifolds. Convergent numerical solutions are found
using this method on manifolds admitting negative Ricci scalar curvature
metrics, i.e. those from the H^3 and the H^2× S^1 geometrization
classes. The method fails to produce solutions, however, on all the manifolds
examined here admitting non-negative Ricci scalar curvatures, i.e. those from
the S^3, S^2× S^1, and the E^3 classes. This study also finds that
the accuracy of the convergent solutions produced by hyperbolic relaxation can
be increased significantly by performing fairly low-cost standard elliptic
solves using the hyperbolic relaxation solutions as initial guesses.
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