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High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension
Journal of Computational Physics ( IF 4.1 ) Pub Date : 2020-10-07 , DOI: 10.1016/j.jcp.2020.109898
Simone Chiocchetti , Ilya Peshkov , Sergey Gavrilyuk , Michael Dumbser

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.



中文翻译:

高阶ADER方案和GLM卷曲清洁,用于具有表面张力的可压缩流的一阶双曲公式

在这项工作中,我们介绍了具有表面张力的两相流弱双曲模型的两个新颖的公式化,最近由Schmidmayer等人提出。在模型中,通过使用新的矢量场而不是标量跟踪器来实现相界的跟踪,因此可以将表面力应力张量直接表示为状态变量的代数函数,而无需计算标量追踪器的梯度。该模型的一个有趣且重要的特征是,该界面场遵守卷曲对合约束,也就是说,要求向量场始终无卷曲。

提出的修改旨在恢复模型的强双曲性,并且与在数值磁流体动力学(MHD)领域中开发的保持散度的数值方法密切相关。第一种策略是基于Godunov在60年代和70年代提出的对称双曲和热力学兼容(SHTC)系统的理论,并产生了一个修正的控制方程组,其中包括一些对称项,类似于Powell等人后来采用的方法。等 在90年代获得了理想的MHD方程。第二种技术是Munz等人提出的双曲广义拉格朗日乘数(GLM)发散清洗方法的扩展。在Maxwell和MHD方程的应用中。

我们使用后验有限体积子单元限制,通过高阶ADER-WENO有限体积和ADER间断Galerkin(DG)方法,解决了所得的非保守双曲偏微分方程(PDE)系统,并进行了一系列有关表面张力为主的流动的数值测试以及冲击驱动的流量。我们还为方程提供了一个新的精确解,展示了时空精度高达10个数量级的方案的收敛性,并研究了双曲率和卷曲约束在计算的长期稳定性中的作用。

更新日期:2020-10-07
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