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We consider a system of nonlinear wave equations with constraints that arisesfrom the Einstein equations of general relativity and describes the geometry ofthe so-called Gowdy symmetric spacetimes on T3. We introduce two numericalmethods, which are based on pseudo-spectral approximation. The first approachrelies on marching in the future time-like direction and toward the coordinatesingularity t=0. The second approach is designed from asymptotic formulas thatare available near this singularity; it evolves the solutions in the pasttimelike direction from "final" data given at t=0. This backward method reliesa novel nonlinear transformation, which allows us to reduce the nonlinearsource terms to simple quadratic products of the unknown variables. Numericalexperiments are presented in various regimes, including cases where "spiky"structures are observed as the coordinate singularity is approached. Theproposed backward strategy leads to a robust numerical method which allows usto accurately simulate the long-time behavior of a large class of Gowdyspacetimes.

Computing Gowdy spacetimes via spectral evolution in future and past directions