High Order ADER Schemes and GLM Curl Cleaning for a First Order Hyperbolic Formulation of Compressible Flow with Surface Tension
High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension Simone Chiocchetti∗a, Ilya Peshkova, Sergey Gavrilyukb, Michael Dumbsera aLaboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy bIUSTI UMR 7343-CNRS, D´epartement de M´ecanique, Aix-Marseille Universit´e,5 rue Enrico Fermi, 13453 Marseille, France Abstract In this work, we introduce two novel reformulations of the weakly hyperbolic model for two-phase flow with surface tension, recently forwarded by Schmidmayer et al. In the model, the tracking of phase boundaries is achieved by using a vector interface field, rather than a scalar tracer, so that the surface-force stress tensor can be expressed directly as an algebraic function of the state variables, without requiring the computation of gradients of the scalar tracer. An interesting and important feature of the model is that this interface field obeys a curl involution constraint, that is, the vector field is required to be curl-free at all times. The proposed modifications are intended to restore the strong hyperbolicity of the model, and are closely related to divergence-preserving numerical approaches developed in the field of numerical magnetohydrody- namics (MHD). The first strategy is based on the theory of Symmetric Hyperbolic and Thermodynamically Compatible (SHTC) systems forwarded by Godunov in the 60s and 70s and yields a modified system of governing equations which includes some symmetrisation terms, in analogy to the approach adopted later by Powell et al in the 90s for the ideal MHD equations.
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