Derivation and characteristics analysis of an acoustics–convection upstream resolution algorithm for the two‐dimensional Euler and Navier–Stokes equations

The first of a two‐paper series, this paper introduces a new decomposition not of the hyperbolic flux vector but of the flux vector Jacobian. The paper then details for the Euler and Navier–Stokes equations an intrinsically infinite directional upstream‐bias formulation that rests on the mathematics and physics of multi‐dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the differential equation level, before the spatial discretization, within a characteristics‐bias governing system. Through a decomposition of the Euler flux divergence into multi‐dimensional acoustics and convection–acoustics components, this characteristics‐bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable‐strength upstreaming that correlates with the spatial distribution of characteristic velocities. Copyright © 2005 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.     

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.     

Runge–Kutta time‐stepping schemes with TVD central differencing for the water hammer equations

In the present study, Runge–Kutta schemes are used to simulate unsteady flow in elastic pipes due to sudden valve closure. The spatial derivatives are discretized using a central difference scheme. Second‐order dissipative terms are added in regions of high gradients while they are switched off in smooth flow regions using a total variation diminishing (TVD) switch. The method is applied to both one‐ and two‐dimensional water hammer formulations. Both laminar and turbulent flow cases are simulated. Different turbulence models are tested including the Baldwin–Lomax and Cebeci–Smith models. The results of the present method are in good agreement with analytical results and with experimental data available in the literature. The two‐dimensional model is shown to predict more accurately the frictional damping of the pressure transient. Moreover, through order of magnitude and dimensional analysis, a non‐dimensional parameter is identified that controls the damping of pressure transients in elastic pipes. Copyright © 2006 John Wiley & Sons, Ltd.     

A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

A space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

Provably stable and time‐accurate extensions of Runge–Kutta schemes for CFD computations on moving grids

Extending fixed‐grid time integration schemes for unsteady CFD applications to moving grids, while formally preserving their numerical stability and time accuracy properties, is a nontrivial task. A general computational framework for constructing stability‐preserving ALE extensions of Eulerian multistep time integration schemes can be found in the literature. A complementary framework for designing accuracy‐preserving ALE extensions of such schemes is also available. However, the application of neither of these two computational frameworks to a multistage method such as a Runge–Kutta (RK) scheme is straightforward. Yet, the RK methods are an important family of explicit and implicit schemes for the approximation of solutions of ordinary differential equations in general and a popular one in CFD applications. This paper presents a methodology for filling this gap. It also applies it to the design of ALE extensions of fixed‐grid explicit and implicit second‐order time‐accurate RK (RK2) methods. To this end, it presents the discrete geometric conservation law associated with ALE RK2 schemes and a method for enforcing it. It also proves, in the context of the nonlinear scalar conservation law, that satisfying this discrete geometric conservation law is a necessary and sufficient condition for a proposed ALE extension of an RK2 scheme to preserve on moving grids the nonlinear stability properties of its fixed‐grid counterpart. All theoretical findings reported in this paper are illustrated with the ALE solution of inviscid and viscous unsteady, nonlinear flow problems associated with vibrations of the AGARD Wing 445.6. Copyright © 2011 John Wiley & Sons, Ltd.     

A hybrid time stepping scheme combining explicit Runge–Kutta with implicit LU‐SGS for Navier–Stokes equations

A hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU‐SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU‐SGS. Copyright © 2015 John Wiley & Sons, Ltd.     

Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder

This paper presents a numerical study of a two‐dimensional time‐dependent flow around a cylinder. Its main objective is to provide accurate reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time. In addition, the accuracy of these values obtained with different time stepping schemes and different finite element methods is studied. Copyright © 2004 John Wiley & Sons, Ltd.     

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

In this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Investigation and application of point implicit Runge–Kutta methods to inviscid flow problems

We derive and investigate point implicit Runge–Kutta methods to significantly improve the convergence rate to approximate steady‐state solutions of inviscid flows. It turns out that the point implicit Runge–Kutta can be interpreted as a preconditioned explicit Runge–Kutta method, where the preconditioner arises naturally as local derivative of the residual function. Moreover, many preconditioners suggested in the literature so far are identified as special case of our general ansatz. Conditions will be formulated such that explicit Runge–Kutta methods with local time stepping are equivalent to point implicit methods. In numerical examples, we will demonstrate the improved convergence rates. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical investigation of a spectral‐type nodal collocation discontinuous Galerkin approximation of the Euler and Navier–Stokes equations

In this paper, we focus on the applicability of spectral‐type collocation discontinuous Galerkin methods to the steady state numerical solution of the inviscid and viscous Navier–Stokes equations on meshes consisting of curved quadrilateral elements. The solution is approximated with piecewise Lagrange polynomials based on both Legendre–Gauss and Legendre–Gauss–Lobatto interpolation nodes. For the sake of computational efficiency, the interpolation nodes can be used also as quadrature points. In this case, however, the effect of the nonlinearities in the equations and/or curved elements leads to aliasing and/or commutation errors that may result in inaccurate or unstable computations. By a thorough numerical testing on a set of well known test cases available in the literature, it is here shown that the two sets of nodes behave very differently, with a clear advantage of the Legendre–Gauss nodes, which always displayed an accurate and robust behaviour in all the test cases considered.Copyright © 2012 John Wiley & Sons, Ltd.     

Fourth‐order finite difference scheme for the incompressible Navier–Stokes equations in a disk

We develop an efficient fourth‐order finite difference method for solving the incompressible Navier–Stokes equations in the vorticity‐stream function formulation on a disk. We use the fourth‐order Runge–Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth‐order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Colocated schemes for the incompressible Navier–Stokes equations on non‐smooth grids for two‐dimensional problems

The accuracy of colocated finite volume schemes for the incompressible Navier–Stokes equations on non‐smooth curvilinear grids is investigated. A frequently used scheme is found to be quite inaccurate on non‐smooth grids. In an attempt to improve the accuracy on such grids, three other schemes are described and tested. Two of these are found to give satisfactory results. Copyright © 2000 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.     

An implicit edge‐based ALE method for the incompressible Navier–Stokes equations

A new finite volume method for the incompressible Navier–Stokes equations, expressed in arbitrary Lagrangian–Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge‐based data structure and assembly procedure which is valid for arbitrary n‐sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi‐stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second‐order‐accurate in time and space for general time‐dependent polygonal meshes. The method is first evaluated using several well‐known unsteady incompressible Navier–Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem. Published in 2003 by John Wiley & Sons, Ltd.     

Two‐level consistent splitting methods based on three corrections for the time‐dependent Navier–Stokes equations

Three kinds of two‐level consistent splitting algorithms for the time‐dependent Navier–Stokes equations are discussed. The basic technique of two‐level type methods for solving the nonlinear problem is first to solve a nonlinear problem in a coarse‐level subspace, then to solve a linear equation in a fine‐level subspace. Hence, the two‐level methods can save a lot of work compared with the one‐level methods. The approaches to linearization are considered based on Stokes, Newton, and Oseen corrections. The stability and convergence demonstrate that the two‐level methods can acquire the optimal accuracy with the proper choice of the coarse and fine mesh scales. Numerical examples show that Stokes correction is the simplest, Newton correction has the best accuracy, while Oseen correction is preferable for the large Reynolds number problems and the long‐time simulations among the three methods. Copyright © 2015 John Wiley & Sons, Ltd.