Abstract
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.
Highlights
Since the advent of modern computers and the first numerical methods for the approximate solution of nonlinear partial differential equations (PDE), the field of computational fluid dynamics (CFD) has been a major driving force for research and new developments in applied mathematics and scientific computing in the last decades, see (e.g., [1,2,3]) for the first finite difference schemes for CFD
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
This paper presents a massively parallel implementation of a novel and versatile family of pressure-based staggered semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics using the Message-Passing Interface (MPI) standard
Summary
Since the advent of modern computers and the first numerical methods for the approximate solution of nonlinear partial differential equations (PDE), the field of computational fluid dynamics (CFD) has been a major driving force for research and new developments in applied mathematics and scientific computing in the last decades, see (e.g., [1,2,3]) for the first finite difference schemes for CFD. The final structure of the matrix-vector product that needs to be evaluated in each CG iteration is similar to the one of an explicit method and can be parallelized as usual at the aid of spatial domain decomposition In this way, the computational cost is shared among all the processors, both for the explicit FV part as well as for the implicit FE part of the hybrid FV/FE scheme. Future work will concern the extension to mixed element unstructured meshes, including prisms, pyramids, and hexahedra, as well as more general polygonal and polyhedral meshes, using e.g., the virtual element method (VEM), see [71,72,73] and references therein Another limitation of the algorithm is the use of a pure MPI parallelization instead of a mixed MPI + OpenMP implementation.
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