The numerical modelling of convection dominated high density ratio two-phase flow poses several challenges, amongst which is resolving the relatively thin shear layer at the interface. To this end we propose a sharp discretisation of the two-velocity model of the two-phase Navier-Stokes (NS) equations. This results in the ability to model the shear layer, rather than resolving it, by allowing for a velocity discontinuity in the direction(s) tangential to the interface.
In this paper we focus our attention on the transport of mass and momentum in the presence of such a velocity discontinuity. We propose a generalisation of the dimensionally unsplit geometric volume of fluid (VOF) method for the advection of the interface in the two-velocity formulation. Sufficient conditions on the construction of donating regions are derived that ensure boundedness of the volume fraction for dimensionally unsplit advection methods. We propose to interpolate the mass fluxes resulting from the dimensionally unsplit geometric VOF method for the advection of the staggered momentum field, resulting in semi-discrete energy conservation. Division of the momentum by the respective mass, to obtain the velocity, is not always well-defined for nearly empty control volumes and therefore care is taken in the construction of the momentum flux interpolant: our proposed flux interpolant guarantees that this division is always well-defined without being unnecessarily dissipative.
Besides the newly proposed two-velocity model we also detail our exactly conservative (mass per phase and total linear momentum) implementation of the one-velocity formulation of the two-phase NS equations, which will be used for comparison.
The discretisation methods are validated using classical time-reversible flow fields, where in this paper the advection is uncoupled from the NS solver, which will be developed in a later paper.
A novel linearized and momentum-preserving Fourier pseudo-spectral scheme for the Rosenau-Korteweg de Vries equation
在本文中，我们设计了一个新的线性化和动量保持傅立叶伪谱格式来解决Rosenau-Korteweg de Vries方程。借助于Fourier准谱方法与有限差分法之间的一种新的半范数等价性，从离散动量守恒律出发，得到了离散$L^{\infty}$范数下数值解的先验界.随后，基于能量方法和数值解的界，我们证明了在不限制网格比的情况下，该格式在离散$L^\infty$ -范数下以阶$O(N^{-s}+τ^2)$收敛，其中$N$是谱方法中使用的配置点数目，$τ$是时间步长.数值结果证实了我们的理论分析。
Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation
Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；College of Science, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院）
On the convergence of iterated penalty methods for structure-preserving discretizations of saddle point problems
University of Oxford（牛津大学）；Charles University（查理大学）；University of Pittsburgh（匹兹堡大学）；U.S. Naval Research Laboratory（美国海军研究实验室）；University of Chicago（芝加哥大学）
Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Technische Universität München（慕尼黑工业大学）
High-order multi-structures-preserving exponential integrators for the derivative nonlinear Schrödinger equation
Faculty of Information Engineering and Automation, Kunming University of Science and Technology（昆明理工大学信息工程与自动化学院）；School of Statistics and Mathematics, Yunnan University of Finance and Economics（云南财经大学统计与数学学院）
Positivity preserving DG schemes for a Boltzmann - Poisson model of electrons in semiconductors in curvilinear momentum coordinates
TU Wien - Institute for Analysis and Scientific Computing（维也纳工业大学分析与科学计算研究所）；The University of Texas at Austin - Institute for Computational Engineering and Sciences & Department of Mathematics（德克萨斯大学奥斯汀分校计算工程与科学研究所及数学系）；Oak Ridge National Lab - Computational and Applied Mathematics Group（橡树岭国家实验室计算与应用数学组）
