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In this paper we propose an extension of the generalized Lagrangianmultiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which wasoriginally conceived for the numerical solution of the Maxwell and MHDequations with divergence-type involutions, to the case of hyperbolic PDEsystems with curl-type involutions. The key idea here is to solve an augmentedPDE system, in which curl errors propagate away via a Maxwell-type evolutionsystem. The new approach is first presented on a simple model problem, in orderto explain the basic ideas. Subsequently, we apply it to a strongly hyperbolicfirst order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein fieldequations of general relativity, which is endowed with 11 curl constraints.Several numerical examples, including the long-time evolution of a stableneutron star in anti-Cowling approximation, are presented in order to show theobtained improvements with respect to the standard formulation without specialtreatment of the curl involution constraints. The main advantages of theproposed GLM approach are its complete independence of the underlying numericalscheme and grid topology and its easy implementation into existing computercodes. However, this flexibility comes at the price of needing to add for eachcurl involution one additional 3 vector plus another scalar in the augmentedsystem for homogeneous curl constraints, and even two additional scalars fornon-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneouscurl involutions, this means that additional 44 evolution quantities need to beadded.

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations