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

Dumbser, Michael and Fambri, Francesco and Gaburro, Elena and Reinarz, Anne (2020) 'On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations.', Journal of computational physics., 404 . p. 109088.


In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. [52], [30], which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions. The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwell-type evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints. The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2020 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:30 October 2020
Date deposited:03 September 2020
Date of first online publication:06 November 2020
Date first made open access:06 November 2021

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