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.

中文翻译：

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Quo vadis， wave？用于直接范德华模拟 （DVS） 的 Dispersive-SUPG


偏微分方程的解以双曲线波和色散波的组合为主，这在多相流中很常见。我们表明，对于这些问题，在不考虑解的色散特性的情况下，应用基于 Streamline-Upwind/Petrov-Galerkin （SUPG） 的经典稳定有限元会导致顺风离散化和不稳定的数值解。为了应对这一挑战，我们提出了 Dispersive-SUPG （D-SUPG） 配方。我们将 Dispersive-SUPG 公式应用于 Korteweg-de Vries 方程和 Direct van der Waals 模拟。数值结果表明，Dispersive-SUPG 是一种高阶、准确和高效的稳定方法，当解以双曲波或色散波为主时，能够产生稳定的结果。最后，我们将所提出的算法应用于二维楔形和三维半球上的空化流，并与理论和实验取得了良好的一致性。