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
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DOI：
10.1016/j.jcp.2020.109898
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发表时间：
2021-01-05
影响因子：
4.1
通讯作者：
Dumbser, Michael
中科院分区：
文献类型：
--
作者：
Chiocchetti, Simone;Peshkov, Ilya;Dumbser, Michael
在这项工作中，我们介绍了Schmidmayer等人最近提出的具有表面张力的两相流弱双曲模型的两个新的重新公式。在该模型中，通过使用新的矢量场而不是标量跟踪器来实现对相边界的跟踪，从而使表面力应力张量可以直接表示为状态变量的代数函数，而不需要计算标量跟踪器的梯度。该模型的一个有趣而重要的特点是，该界面场服从旋度对合约束，即要求矢量场在任何时候都是无旋度的。所提出的修改旨在恢复模型的强双曲性，并且与数值磁流体领域中发展的保散数值方法密切相关。第一种策略基于Godunov在60年代和70年代提出的对称双曲热力学相容(SHTC)系统理论，与Powell等人后来采用的方法类似，得到了一个修改的控制方程组，其中包括一些对称性项。在90年代的理想MHD方程。第二种方法是Munz等人提出的双曲广义拉格朗日乘子(GLM)散度清除方法的扩展。在Maxwell方程和MHD方程的应用中，我们用高阶Ader-Weno有限体积法和Ader间断Galerkin(DG)方法分别求解了高阶双曲偏微分方程组和Ader间断Galerkin(DG)方法，并进行了一系列关于表面张力主导流动和激波驱动流动的数值试验。我们还给出了方程的一个新的精确解，证明了精度可达10阶的格式在空间和时间上的收敛，并研究了双曲性和旋度约束在计算的长期稳定性中的作用。(C)2020 Elsevier Inc.保留所有权利。
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 new vector 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 magnetohydrodynamics (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. The second technique is an extension of the hyperbolic Generalized Lagrangian Multiplier (GLM) divergence cleaning approach, forwarded by Munz et al. in applications to the Maxwell and MHD equations.We solve the resulting nonconservative hyperbolic partial differential equation (PDE) systems with high order ADER-WENO Finite Volume and ADER Discontinuous Galerkin (DG) methods with a posteriori Finite Volume subcell limiting and carry out a set of numerical tests concerning flows dominated by surface tension as well as shock-driven flows. We also provide a new exact solution to the equations, show convergence of the schemes for orders of accuracy up to ten in space and time, and investigate the role of hyperbolicity and of curl constraints in the long-term stability of the computations. (C) 2020 Elsevier Inc. All rights reserved.