Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1375-1408, January 2020.
We present and discuss a novel approach to deal with conservation properties for the simulation of nonlinear complex porous media flows in the presence of the following: (1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model and (2) multiscale wave structures resulting from interactions of shock waves and rarefaction from the nonlinear hyperbolic-transport model. For the pressure-velocity Darcy flow problem, we revisit a recent high-order and volumetric residual-based Lagrange multipliers saddle point problem to impose local mass conservation on convex polygons. We clarify and improve conservation properties on applications. For the hyperbolic-transport problem we introduce a new locally conservative Lagrangian--Eulerian finite volume method. For the purpose of this work, we recast our method within the Crandall and Majda treatment of the stability and convergence properties of conservation-form, monotone difference, in which the scheme converges to the physical weak solution satisfying the entropy condition. This multiscale coupling approach was applied to several nontrivial examples to show that we are computing qualitatively correct reference solutions. We combine these procedures for the simulation of the fundamental two-phase flow problem with high-contrast multiscale porous medium, but recalling state-of-the-art paradigms on the notion of solution in related multiscale applications. This is a first step to deal with out-of-reach multiscale systems with traditional techniques. We provide robust numerical examples for verifying the theory and illustrating the capabilities of the approach being presented.

中文翻译：

---

复流中达西流与双曲输运的多尺度耦合和模拟中的守恒性质

2020年1月，《多尺度建模与仿真》，第18卷，第4期，第1375-1408页。
我们提出并讨论了一种新颖的方法来处理守恒性质，以模拟以下情况下的非线性复杂多孔介质流：（1）椭圆压力-速度和岩石地质模型中出现的多尺度非均质结构，以及（ 2）非线性双曲传输模型中冲击波和稀疏性相互作用产生的多尺度波结构。对于压力-速度达西流动问题，我们重新研究了最近的基于高阶和体积残差的拉格朗日乘子鞍点问题，以对凸多边形施加局部质量守恒。我们阐明并改善应用程序的保护特性。对于双曲输运问题，我们引入了一种新的局部保守拉格朗日—欧拉有限体积方法。为了这项工作，我们在Crandall和Majda处理中重整了守恒形式，单调差分形式的稳定性和收敛性的方法，其中该方案收敛到满足熵条件的物理弱解。这种多尺度耦合方法已应用于几个非平凡的例子，以表明我们正在计算定性正确的参考解决方案。我们将这些程序与高对比度多尺度多孔介质的基本两相流问题模拟相结合，但回顾了有关多尺度应用中解决方案概念的最新范例。这是用传统技术处理超范围多尺度系统的第一步。我们提供了可靠的数值示例来验证理论并说明所提出方法的功能。