Invited Lecture
On the treatment of curl involutions in Newtonian continuum mechanics and general relativity
University of Trento, Italy
Abstract
In this talk we present several first order hyperbolic PDE models from Newtonian continuum mechanics and from general relativity that are endowed with curl constraints. In particular, we consider the recent first order hyperbolic system of compressible multi-phase flows with surface tension proposed by Gavrilyuk et al., the first order hyperbolic reformulation of the nonlinear defocusing Schrödinger equation and a new first order hyperbolic reformulation of the Einstein field equations of general relativity (FO-CCZ4). In all cases, evolution equations for auxiliary gradient fields are added to the original PDE system, leading to the natural constraint that the curl of the auxiliary field must remain zero for all times if it was initially zero. In many cases, the resulting first order hyperbolic system is only weakly hyperbolic, which poses several challenges from a numerical point of view. In this talk we present a novel generalization of the successful generalized Lagragian multiplier (GLM) approach of Munz et al. to the treatment of PDE with curl constraints. The proposed GLM curl cleaning approach is conservative and allows to restore strong hyperbolicity of the system. We show several numerical examples in the context of Newtonian mechanics and in general relativity.