Quo vadis, wave? Dispersive-SUPG for direct van der Waals simulation (DVS)

Partial differential equations whose solution is dominated by a combination of hyperbolic and dispersive waves are common in multiphase flows. We show that for these problems, the application of classical stabilized finite elements based on Streamline-Upwind/Petrov–Galerkin (SUPG) without accounting for the dispersive features of the solution leads to a downwind discretization and an unstable numerical solution. To address this challenge, we propose the Dispersive-SUPG (D-SUPG) formulation. We apply the Dispersive-SUPG formulation to the Korteweg–de Vries equation and Direct van der Waals Simulations. Numerical results show that Dispersive-SUPG is a high-order accurate and efficient stabilized method, capable of producing stable results when the solution is dominated by either hyperbolic or dispersive waves. We finally applied the proposed algorithm to study cavitating flow over a 2D wedge and a 3D hemisphere and obtained good agreement with theory and experiments.