Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Practical aspects of p‐multigrid discontinuous Galerkin solver for steady and unsteady RANS simulations

Efficient and robust p‐multigrid solvers are presented for solving the system arising from high‐order discontinuous Galerkin discretizations of the compressible Reynolds‐Averaged Navier–Stokes (RANS) equations. Two types of multigrid methods and a multigrid preconditioned Newton–Krylov method are investigated, and both steady and unsteady algorithms are considered in this paper. For steady algorithms, a new strategy is introduced to determine the CFL number, which has been proved to be critical in achieving the effective and stable convergence for p‐multigrid methods. We also suggest a modified smoothing technique to further improve the efficiency of the algorithms. For unsteady algorithms, special attention has been paid to the cycling strategy and the full multigrid technique, and we point out a significant difference on the parameter selection for unsteady computations. The capabilities of the resulted solvers have been examined by performing steady and unsteady RANS simulations. Comparative assessment in terms of efficiency, robustness, and memory consumption are carried out for all solvers. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical flux functions for Reynolds‐averaged Navier–Stokes and kω turbulence model computations with a line‐preconditioned p‐multigrid discontinuous Galerkin solver

We present an eigen‐decomposition of the quasi‐linear convective flux formulation of the completely coupled Reynolds‐averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p‐multigrid algorithm. To this end, we formulate a framework with a backward‐Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix‐free implementations and memory‐lean line‐Jacobi preconditioners and compare the effects of some parameter choices. In particular, p‐multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.     

A multigrid method with higher‐order discretization schemes

The implementation of the multigrid method into the SIMPLE algorithm presents interesting aspects concerning the mass fluxes conservation on coarser grids, the k–ε turbulence model and the higher‐order discretization schemes. Higher‐order discretization schemes for the convection terms are increasingly used in order to guarantee accuracy in demanding engineering applications. However, when used in single‐grid algorithms, their convergence is considerably slower compared with the first‐order schemes. Unbounded higher‐order schemes offer maximum accuracy, but quite often they do not converge due to their oscillatory behaviour. This paper demonstrates the dual function of the multigrid method: reduction of CPU time and stabilization of the iterating procedure, making it possible to perform computations with the third‐order accurate QUICK scheme in all cases. The method is applied to the calculation of two‐ and three‐dimensional flows with or without turbulence modelling. The results show that the convergence rate of the present algorithm does not deteriorate when QUICK is used and that, if applied on complex engineering cases, large gains in computational time can be achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows

High‐speed compressible turbulent flows typically contain discontinuities and have been widely modeled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1.5 and Re_τ = 222 is conducted using OpenSBLI, a high‐order finite difference computational fluid dynamics framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against the WALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The sixth‐order adaptive TENO scheme is found to produce comparable results to those obtained with nondissipative fourth‐ and sixth‐order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.     

Monolithic Newton‐multigrid solution techniques for incompressible nonlinear flow models

We present special Newton‐multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB‐stable Q₂P₁ element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well‐known flow around cylinder benchmark configuration for generalized Newtonian as well as non‐Newtonian flows including non‐isothermal, shear/pressure dependent and viscoelastic effects.Copyright © 2012 John Wiley & Sons, Ltd.     

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.     

k‐Version of finite element method in 2‐D polymer flows: Oldroyd‐B constitutive model

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process of two‐dimensional polymer flows utilizing Oldroyd‐B constitutive model. Alternate forms of the choices of dependent variables in the governing differential equations (GDEs) are considered and is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious over others. It is shown that: (a) since, the differential operator in the GDEs is non‐linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non‐symmetric and hence may have partial or completely complex basis and thus the resulting computational processes may be spurious. (b) Since the VC of the VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches always have possibility of spurious solutions. (c) Least squares process utilizing GDEs in u, v, p, τ (strong form of the GDEs) variables (as well as others) is variationally consistent. The coefficient matrices are always symmetric and positive definite and hence always have a real basis and thus naturally yield computational processes that are free of spurious solutions. (d) The theoretical solution of the GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solution are preserved, undoubtedly requires local approximations in higher order scalar product spaces . (e) LSP with local approximations in spaces provide an incomparable mathematical and computational framework in which it is possible to preserve desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of the Oldroyd‐B model or lack of it is examined for both model problems for increasing De. A mathematical idealization of the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h_d→0. For both model problems shear rate is examined in the flow domain to establish validity of the Oldroyd‐B constitutive model. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

The reduced‐order hybrid Monte Carlo sampling smoother

Hybrid Monte Carlo sampling smoother is a fully non‐Gaussian four‐dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive owing to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the hybrid Monte Carlo sampling smoother based on reduced‐order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow‐water equations model on Cartesian coordinates. The results reveal that the reduced‐order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full‐order formulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of pressure‐ and density‐based methods for different flow speeds

In this paper, pressure‐based and density‐based methods are studied at different flow speeds. The methods are intended for steady flows, and the goal is to find as general an approach as possible to cover different Mach number regimes. The solution methods are based on a finite‐volume approach. Various forms of inviscid fluxes are applied and connected with either a pressure‐based or density‐based implicit solution. For this purpose, a new pressure‐correction method is developed that can be applied for incompressible and for compressible flows. Another option is a standard density‐based approximate factorization method. In both cases, a convergence is accelerated with a Full Approximation Scheme (FAS) multigrid approach. Sample problems in the range of Ma = 0…6 are simulated using different approaches, and their efficiency and accuracy are compared. On the basis of the quality of the solutions, recommendations are made. © 2015 The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons Ltd.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Preconditioned Krylov subspace methods used in solving two‐dimensional transient two‐phase flows

This paper investigates the performance of preconditioned Krylov subspace methods used in a previously presented two‐fluid model developed for the simulation of separated and intermittent gas–liquid flows. The two‐fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two‐step semi‐implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi‐CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, τ) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions

In this paper, a new family of high‐order relaxation methods is constructed. These methods combine general higher‐order reconstruction for spatial discretization and higher order implicit‐explicit schemes or TVD Runge–Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow‐water equations in both one and two space dimensions on flat and non‐flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Development of temporal and spatial high‐order schemes for two‐fluid seven‐equation two‐pressure model and its applications in two‐phase flow benchmark problems

Current existing main nuclear thermal‐hydraulics (T‐H) system analysis codes, such as RALAP5, TRACE, and CATHARE, play a crucial role in the nuclear engineering field for the design and safety analysis of nuclear reactor systems. However, two‐fluid model used in these T‐H system analysis codes is ill posed, easily leading to numerical oscillations, and the classical first‐order methods for temporal and special discretization are widely employed for numerical simulations, yielding excessive numerical diffusion. Two‐fluid seven‐equation two‐pressure model is of particular interest due to the inherent well‐posed advantage. Moreover, high‐order accuracy schemes have also attracted great attention to overcome the challenge of serious numerical diffusion induced by low‐order time and space schemes for accurately simulating nuclear T‐H problems. In this paper, the semi‐implicit solution algorithm with high‐order accuracy in space and time is developed for this well‐posed two‐fluid model and the robustness and accuracy are verified and assessed against several important two‐phase flow benchmark tests in the nuclear engineering T‐H field, which include two linear advection problems, the oscillation problem of the liquid column, the Ransom water faucet problem, the reversed water faucet problem, and the two‐phase shock tube problem. The following conclusions are achieved. (1) The proposed semi‐implicit solution algorithm is robust in solving two‐phase flows, even for fast transients and discontinuous solutions. (2) High‐order schemes in both time and space could prevent excessive numerical diffusion effectively and the numerical simulation results are more accurate than those of first‐order time and space schemes, which demonstrates the advantage of using high‐order schemes.     

An hp‐adaptive unstructured finite volume solver for compressible flows

The idea of hp‐adaptation, which has originally been developed for compact schemes (such as finite element methods), suggests an adaptation scheme using a mixture of mesh refinement and order enrichment based on the smoothness of the solution to obtain an accurate solution efficiently. In this paper, we develop an hp‐adaptation framework for unstructured finite volume methods using residual‐based and adjoint‐based error indicators. For the residual‐based error indicator, we use a higher‐order discrete operator to estimate the truncation error, whereas this estimate is weighted by the solution of the discrete adjoint problem for an output of interest to form the adaptation indicator for adjoint‐based adaptations. We perform our adaptation by local subdivision of cells with nonconforming interfaces allowed and local reconstruction of higher‐order polynomials for solution approximations. We present our results for two‐dimensional compressible flow problems including subsonic inviscid, transonic inviscid, and subsonic laminar flow around the NACA 0012 airfoil and also turbulent flow over a flat plate. Our numerical results suggest the efficiency and accuracy advantages of adjoint‐based hp‐adaptations over uniform refinement and also over residual‐based adaptation for flows with and without singularities.     

Multiple semi‐coarsened multigrid method with application to large eddy simulation

The Multiple Semi‐coarsened Grid (MSG) multigrid method of Mulder (J. Comput. Phys. 1989; 83 :303–323) is developed as a solver for fully implicit discretizations of the time‐dependent incompressible Navier–Stokes equations. The method is combined with the Symmetric Coupled Gauss–Seidel (SCGS) smoother of Vanka (Comput. Methods Appl. Mech. Eng. 1986; 55 :321–338) and its robustness demonstrated by performing a number of large‐eddy simulations, including bypass transition on a flat plate and the turbulent thermally‐driven cavity flow. The method is consistently able to reduce the non‐linear residual by 5 orders of magnitude in 40–80 work units for problems with significant and varying coefficient anisotropy. Some discussion of the parallel implementation of the method is also included. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.