In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

中文翻译：

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用于计算流体动力学的大规模并行混合有限体积/有限元方案

在本文中，我们为计算流体动力学 (CFD) 提出了一系列新的半隐式混合有限体积/有限元方案，特别是对于不可压缩和可压缩 Navier-Stokes 方程的近似解，以及浅层二维和三个空间维度中交错非结构化网格上的水方程。该方法的主要特点是使用基于边缘/基于面的交错双网格在显式高分辨率 Godunov 型有限体积方案的帮助下离散化非线性对流项，而压力项使用隐式离散化原始单纯形网格上的经典连续拉格朗日有限元。由此产生的压力系统是对称正定的，因此可以在经典 Krylov 子空间方法的帮助下非常有效地求解，例如无矩阵共轭梯度法。对于可压缩的 Navier-Stokes 方程，该方案在方程的低马赫数限制中通过构造渐近保持，因此获得了不可压缩方程的一致混合 FV/FE 方法。算法的所有部分都可以有效地并行化，即显式有限体积步骤以及隐式压力求解器中的矩阵向​​量乘积。关于并行实现，我们采用消息传递接口 (MPI) 标准并结合基于免费软件包 METIS 的空间域分解。为了展示所提议方案的多功能性，我们提出了广泛的应用，从环境和地球物理流开始，例如堤坝问题和自然对流，从湍流不可压缩流到具有冲击波的高马赫数可压缩流的直接数值模拟。在所有情况下都实现了与精确解析、数值或实验参考解决方案的极好一致性。大多数模拟都是在数千个 CPU 内核上以数百万个自由度运行的。我们展示了应用于 3D 不可压缩 Navier-Stokes 方程的混合 FV/FE 方案的强大缩放结果，使用数百万个自由度和多达 4096 个 CPU 内核。本文中显示的最大模拟是著名的 3D Taylor-Green 涡旋基准测试，该基准测试在 32,768 个 CPU 内核上的 6.71 亿个四面体元素上运行，清楚地表明所提出的算法适用于解决现代大规模并行分布式内存上的大型 CFD 问题超级计算机。