High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension

• Two novel strongly hyperbolic reformulations of a model for surface tension. Godunov-Powell-type formulation PDE with curl-type involutions. Generalized Lagrangian multiplier (GLM) approach Long-time simulations stationary and oscillating bubbles shock-bubble interaction. High order ADER discontinuous Galerkin schemes posteriori subcell finite volume limiter. In this work, we introduce two the weakly two-phase flow tension, recently forwarded by Schmidmayer et al. model, tracking phase boundaries is achieved using new vector field , rather than scalar tracer, so that surface-force stress tensor can be expressed directly as an algebraic function state variables, without requiring computation gradients tracer. An interesting important feature interface obeys curl involution constraint is, required to curl-free at all times. The proposed modifications are intended restore strong hyperbolicity closely related divergence-preserving numerical approaches developed in magnetohydrodynamics (MHD). first strategy based on theory Symmetric Hyperbolic Thermodynamically Compatible (SHTC) systems Godunov 60s 70s yields modified system governing equations which includes some symmetrisation terms, analogy adopted later Powell 90s ideal MHD equations. second technique extension Multiplier divergence cleaning approach, Munz applications Maxwell We solve resulting nonconservative partial differential equation (PDE) high ADER-WENO Finite Volume Discontinuous (DG) methods limiting carry out set tests concerning flows dominated tension well shock-driven flows. also provide exact solution equations, show convergence orders accuracy up ten space time, investigate role constraints long-term stability computations.