Linear stability analysis shows the algebraic-hyperbolic formulation of constraint equations is unstable near FLRW but can be stable for Gowdy spacetimes and subclasses with Fourier-based methods.
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Numerical stability of the Hyperbolic Formulation of the Constraint equations for $\mathbb{T}^3$ cosmological space-times
Linear stability analysis shows the algebraic-hyperbolic formulation of constraint equations is unstable near FLRW but can be stable for Gowdy spacetimes and subclasses with Fourier-based methods.