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1. A consistent adaptive level set framework for incompressible two-phase flows with high density ratios and high Reynolds numbers

   Zeng, Y; Liu, H; Gao, Q; Almgren, A; Bhalla, A; Shen, L (January 2023, Journal of computational physics)

   We develop a consistent adaptive framework in a multilevel collocated grid layout for simulating two-phase flows with adaptive mesh refinement (AMR). The conservative mo-mentum equations and the mass equation are solved in the present consistent framework. This consistent mass and momentum transport treatment greatly improves the accuracy and robustness for simulating two-phase flows with a high density ratio and high Reynolds number. The interface capturing level set method is coupled with the conservative form of the Navier–Stokes equations, and the multilevel reinitialization technique is applied for mass conservation. This adaptive framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method to decouple the data ad-vancement on each level. The accuracy and robustness of the framework are validated by a variety of canonical two-phase flow problems. We demonstrate that the consistent scheme results in a numerically stable solution in flows with high density ratios(up to 106) and high Reynolds numbers(up to 106), while the inconsistent scheme exhibits non-physical fluid behaviors in these tests. Furthermore, it is shown that the subcycling and non-subcycling methods provide consistent results and that both of them can accurately resolve the interfaces of the two-phase flows with surface tension effects. Finally, a 3D breaking wave problem is simulated to show the efficiency and significant speedup of the proposed framework using AMR. 

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2. A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

   https://doi.org/https://doi.org/10.1016/j.cpc.2022.108501  

   Makrand A. Khanwale; Kumar Saurabh; Milinda Fernando; Victor M.Calo; Hari Sundar; James A.Rossmanith; Baskar Ganapathysubramanian (November 2022, Computer physics communications)

   We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation. 

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3. Von Neumann Stable, Implicit, High Order, Finite Volume WENO Schemes

   https://doi.org/10.2118/193817-MS  

   Arbogast, Todd; Huang, Chieh-Sen; Zhao, Xikai (April 2019, SPE Reservoir Simulation Conference 2019)

   Simulation of flow and transport in petroleum reservoirs involves solving coupled systems of advection-diffusion-reaction equations with nonlinear flux functions, diffusion coefficients, and reactions/wells. It is important to develop numerical schemes that can approximate all three processes at once, and to high order, so that the physics can be well resolved. In this paper, we propose an approach based on high order, finite volume, implicit, Weighted Essentially NonOscillatory (iWENO) schemes. The resulting schemes are locally mass conservative and, being implicit, suited to systems of advection-diffusion-reaction equations. Moreover, our approach gives unconditionally L-stable schemes for smooth solutions to the linear advection-diffusion-reaction equation in the sense of a von Neumann stability analysis. To illustrate our approach, we develop a third order iWENO scheme for the saturation equation of two-phase flow in porous media in two space dimensions. The keys to high order accuracy are to use WENO reconstruction in space (which handles shocks and steep fronts) combined with a two-stage Radau-IIA Runge-Kutta time integrator. The saturation is approximated by its averages over the mesh elements at the current time level and at two future time levels; therefore, the scheme uses two unknowns per grid block per variable, independent of the spatial dimension. This makes the scheme fairly computationally efficient, both because reconstructions make use of local information that can fit in cache memory, and because the global system has about as small a number of degrees of freedom as possible. The scheme is relatively simple to implement, high order accurate, maintains local mass conservation, applies to general computational meshes, and appears to be robust. Preliminary computational tests show the potential of the scheme to handle advection-diffusion-reaction processes on meshes of quadrilateral gridblocks, and to do so to high order accuracy using relatively long time steps. The new scheme can be viewed as a generalization of standard cell-centered finite volume (or finite difference) methods. It achieves high order in both space and time, and it incorporates WENO slope limiting. 

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4. Comparison of adaptive mesh refinement techniques for numerical weather prediction

   Abdi, Daniel S; Almgren, Ann; Giraldo, Francis X; Jankov, Isidora (April 2024, arXiv but under review in Monthly Weather Review)

   This paper examines the application of adaptive mesh refinement (AMR) in the field of numerical weather prediction (NWP). We implement and assess two distinct AMR approaches and evaluate their performance through standard NWP benchmarks. In both cases, we solve the fully compressible Euler equations, fundamental to many non-hydrostatic weather models. The first approach utilizes oct-tree cell-based mesh refinement coupled with a high-order discontinuous Galerkin method for spatial discretization. In the second approach, we employ level-based AMR with the finite difference method. Our study provides insights into the accuracy and benefits of employing these AMR methodologies for the multi-scale problem of NWP. Additionally, we explore essential properties including their impact on mass and energy conservation. Moreover, we present and evaluate an AMR solution transfer strategy for the tree-based AMR approach that is simple to implement, memory-efficient, and ensures conservation for both flow in the box and sphere. Furthermore, we discuss scalability, performance portability, and the practical utility of the AMR methodology within an NWP framework -- crucial considerations in selecting an AMR approach. The current de facto standard for mesh refinement in NWP employs a relatively simplistic approach of static nested grids, either within a general circulation model or a separately operated regional model with loose one-way synchronization. It is our hope that this study will stimulate further interest in the adoption of AMR frameworks like AMReX in NWP. These frameworks offer a triple advantage: a robust dynamic AMR for tracking localized and consequential features such as tropical cyclones, extreme scalability, and performance portability. 

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5. Validation of an OpenFOAM®-based solver for the Euler equations with benchmarks for mesoscale atmospheric modeling

   https://doi.org/10.1063/5.0147457  

   Girfoglio, Michele; Quaini, Annalisa; Rozza, Gianluigi (May 2023, AIP Advances)

   Within OpenFOAM, we develop a pressure-based solver for the Euler equations written in conservative form using density, momentum, and total energy as variables. Under simplifying assumptions, these equations are used to describe non-hydrostatic atmospheric flow. For the stabilization of the Euler equations and to capture sub-grid processes, we consider two Large Eddy Simulation models: the classical Smagorinsky model and the one equation eddy-viscosity model. To achieve high computational efficiency, our solver uses a splitting scheme that decouples the computation of each variable. The numerical results obtained with our solver are validated against numerical data available in the literature for two classical benchmarks: the rising thermal bubble and the density current. Through qualitative and quantitative comparisons, we show that our approach is accurate. This paper is meant to lay the foundation for a new open-source package specifically created for the quick assessment of new computational approaches for the simulation of atmospheric flows at the mesoscale level. 

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