Algorithms for studying transitions and instabilities in incompressible flows typically require the solution of linear systems with the full Jacobian matrix. Other popular approaches, like gradient-based design optimization and fully implicit time integration, also require very robust solvers for this type of linear system. We present a parallel fully coupled multilevel incomplete factorization preconditioner for the 3D stationary incompressible Navier-Stokes equations on a structured grid. The algorithm and software are based on the robust two-level method developed by Wubs and Thies. In this paper, we identify some of the weak spots of the two-level scheme and propose remedies such as a different domain partitioning and recursive application of the method. We apply the method to the well-known 3D lid-driven cavity benchmark problem, and demonstrate its superior robustness by comparing with a segregated SIMPLE-type preconditioner.
The incompressible Navier-Stokes equations are solved in a channel, using a Discontinuous Galerkin method over staggered grids. The resulting linear systems are studied both in terms of the structure and in terms of the spectral features of the related coefficient matrices. In fact, the resulting matrices are of block type, each block showing Toeplitz-like, band, and tensor structure at the same time. Using this rich matrix-theoretic information and the Toeplitz, Generalized Locally Toeplitz technology, a quite complete spectral analysis is presented, with the target of designing and analyzing…
A new high order accurate semi-implicit space-time Discontinuous Galerkin method on staggered grids, for the simulation of viscous incompressible flows on two-dimensional domains is presented. The designed scheme is of the Arbitrary Lagrangian Eulerian type, which is suitable to work on fixed as well as on moving meshes. In our space-time formulation, by expressing the numerical solution in terms of piecewise space-time polynomials, an arbitrary high order of accuracy in time is achieved through a simple and efficient method of Picard iterations. For the dual mesh, the basis functions consist …
A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of d…
A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier-Stokes equations using a staggered-variable arrangement
Institute of Science Tokyo（科学技术研究所）；Kobe University（神户大学）
We present a scalable block preconditioning strategy for the trace system coming from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper block), for which a system algebraic multigrid (AMG) is used for the approx…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine（纽约大学医学院心脏病学莱昂·H·查尼分部）
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes
We present a novel fully implicit hybrid finite volume/finite element method for incompressible flows. Following previous works on semi-implicit hybrid FV/FE schemes, the incompressible Navier-Stokes equations are split into a pressure and a transport-diffusion subsystem. The first of them can be seen as a Poisson type problem and is thus solved efficiently using classical continuous Lagrange finite elements. On the other hand, finite volume methods are employed to solve the convective subsystem, in combination with Crouzeix-Raviart finite elements for the discretization of the viscous stress …
A Convergent Staggered Scheme for the Variable Density Incompressible Navier-Stokes Equations
Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（法国核辐射防护与核安全研究院）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学，法国国家科学研究中心联合研究单位5208，里昂第一大学，卡米耶·若尔当研究所）
In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretiz…
We propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [TD14], in which the spatial domain is discretized using a face-based staggered unstructured mesh. For the computa…
We present a non-nested multilevel algorithm for solving the Poisson equation discretized at scattered points using polyharmonic radial basis function (PHS-RBF) interpolations. We append polynomials to the radial basis functions to achieve exponential convergence of discretization errors. The interpolations are performed over local clouds of points and the Poisson equation is collocated at each of the scattered points, resulting in a sparse set of discrete equations for the unkown variables. To solve this set of equations, we have developed a non-nested multilevel algorithm utilizing multiple …
A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggere…
A novel pressure-free two-fluid model formulation is proposed for the simulation of one-dimensional incompressible multiphase flow in pipelines and channels. The model is obtained by simultaneously eliminating the volume constraint and the pressure from the widely used two-fluid model (TFM). The resulting `pressure-free two-fluid model' (PF-TFM) has a number of attractive features: (i) it features four evolution equations (without additional constraints) that can be solved very quickly with explicit time integration methods; (ii) it keeps the conservation properties of the original two-fluid m…
Computations of incompressible flows with velocity boundary conditions require solution of a Poisson equation for pressure with all Neumann boundary conditions. Discretization of such a Poisson equation results in a rank-deficient matrix of coefficients. When a non-conservative discretization method such as finite difference, finite element, or spectral scheme is used, such a matrix also generates an inconsistency which makes the residuals in the iterative solution to saturate at a threshold level that depends on the spatial resolution and order of the discretization scheme. In this paper, we …
Low Mach number limit of some staggered schemes for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学、法国国家科学研究中心、马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与核安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学、法国国家科学研究中心、里昂第一大学、卡米尔·若尔当研究所）
Mass Conservative and Energy Stable Finite Difference Methods for the Quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive Variable and Projection-Type Schemes
University of California, Irvine（加州大学尔湾分校）；University of Dundee（邓迪大学）；University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）
We address here the discretization of the momentum convection operator for fluid flow simulations on 2D triangular and quadrangular meshes and 3D polyhedral meshes containing hexahedra, tetrahedra, prisms and pyramids. The finite volume scheme that we use for the full Euler equations is based on a staggered discretization: the density unknowns are associated with a primal mesh, whereas the velocity unknowns are associated with a "fictive" dual mesh. Accordingly, the convection operator of the mass balance equation is derived on the primal mesh, while the the convection operator of the momentum…
In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in Bermúdez et al. 2014 and Busto et al. 2018 for viscous incompressible flows is extended here to solve the compressible Navier-Stokes equations. Instead of considering the classical system including the energy conservation equation, we replace it by the pressure evolution equation written in non-conservative form. To ease the discretization of complex spatial d…
This study proposes an algorithm for modeling compressible flows in spherical shells in nearly incompressible and weakly compressible regimes based on an implicit direction splitting approach. The method retains theoretically expected convergence rates and remains stable for extremely small values of the characteristic Mach number. The staggered spatial discretization on the MAC stencil, commonly used in numerical methods for incompressible Navier-Stokes equations, was found to be convenient for the discretization of the compressible Navier-Stokes equations written in the non-conservative form…
In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been largely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals but with very marginal theoretical results. In this paper, we propose a general two-grid convergence analysis proving an optimal convergence rate independent of the matrix s…
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室计算科学与工程中心）；Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine（纽约大学医学院莱昂·H·查尼心脏病学分部）
