Spatial accuracy and performance of a mixed‐order,explicit multi‐stage method for unsteady flows

We assess the spatial accuracy and performance of a mixed‐order, explicit multi‐stage method in which an inexpensive low‐order scheme is used for the initial stages, and a more expensive high‐order scheme is used for the final stage only. Compared with the use of a high‐order scheme for all stages, we observe that the mixed‐order scheme achieves comparable accuracy and convergence while providing a speed‐up of a factor of two on mesh sizes of O(10⁶ ? 10⁷) tetrahedron. For calculations with significant adaptive mesh refinement, a more modest speed‐up of 30% is obtained. Published 2012. This article is a US Government work and is in the public domain in the USA.     

A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

We propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.     

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Numerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd.     

Fifth‐order Hermitian schemes for computational linear aeroacoustics

We develop a class of fifth‐order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth‐order accuracy. A one‐dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy–Kovaleskaya (the ‘Δ‐P5 scheme’); the second one relies on a semi‐discrete form and evolves in time the discrete unknowns by using a five‐stage Runge–Kutta method (the ‘RGK‐P5 scheme’). Although the RGK‐P5 scheme shares the same local spatial interpolator with the Δ‐P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi‐dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210 :133–170; J. Comput. Phys. 2006; 217 :530–562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ‘reconstruction stage’, we define a fifth‐order local spatial interpolator based on an upwind stencil. In the ‘decomposition stage’, we decompose the time derivatives into simple wave contributions. In the ‘evolution stage’, we use these fluctuations to update either by a Cauchy–Kovaleskaya procedure or by a five‐stage Runge–Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ‘evolution stage’, two fifth‐order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK‐P5 appears as the best compromise between simplicity and accuracy, while the Δ‐P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi‐size‐mesh multi‐time‐step algorithm for noise computation on curvilinear meshes

Aeroacoustic problems are often multi‐scale and a zonal refinement technique is thus desirable to reduce computational effort while preserving low dissipation and low dispersion errors from the numerical scheme. For that purpose, the multi‐size‐mesh multi‐time‐step algorithm of Tam and Kurbatskii [AIAA Journal, 2000, 38 (8), p. 1331–1339] allows changes by a factor of two between adjacent blocks, accompanied by a doubling in the time step. This local time stepping avoids wasting calculation time, which would result from imposing a unique time step dictated by the smallest grid size for explicit time marching. In the present study, the multi‐size‐mesh multi‐time‐step method is extended to general curvilinear grids by using a suitable coordinate transformation and by performing the necessary interpolations directly in the physical space due to multidimensional interpolations combining order constraints and optimization in the wave number space. A particular attention is paid to the properties of the Adams–Bashforth schemes used for time marching. The optimization of the coefficients by minimizing an error in the wave number space rather than satisfying a formal order is shown to be inefficient for Adams–Bashforth schemes. The accuracy of the extended multi‐size‐mesh multi‐time‐step algorithm is first demonstrated for acoustic propagation on a sinusoidal grid and for a computation of laminar trailing edge noise. In the latter test‐case, the mesh doubling is close to the airfoil and the vortical structures are crossing the doubling interface without affecting the quality of the radiated field. The applicability of the algorithm in three dimensions is eventually demonstrated by computing tonal noise from a moderate Reynolds number flow over an airfoil. Copyright © 2013 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Verification of a mixed high‐order accurate DNS code for laminar turbulent transition by the method of manufactured solutions

This paper presents results on a verification test of a Direct Numerical Simulation code of mixed high‐order of accuracy using the method of manufactured solutions (MMS). This test is based on the formulation of an analytical solution for the Navier–Stokes equations modified by the addition of a source term. The present numerical code was aimed at simulating the temporal evolution of instability waves in a plane Poiseuille flow. The governing equations were solved in a vorticity–velocity formulation for a two‐dimensional incompressible flow. The code employed two different numerical schemes. One used mixed high‐order compact and non‐compact finite‐differences from fourth‐order to sixth‐order of accuracy. The other scheme used spectral methods instead of finite‐difference methods for the streamwise direction, which was periodic. In the present test, particular attention was paid to the boundary conditions of the physical problem of interest. Indeed, the verification procedure using MMS can be more demanding than the often used comparison with Linear Stability Theory. That is particularly because in the latter test no attention is paid to the nonlinear terms. For the present verification test, it was possible to manufacture an analytical solution that reproduced some aspects of an instability wave in a nonlinear stage. Although the results of the verification by MMS for this mixed‐order numerical scheme had to be interpreted with care, the test was very useful as it gave confidence that the code was free of programming errors. Copyright © 2009 John Wiley & Sons, Ltd.     

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

Compact high‐resolution algorithms for time‐dependent advection on unstructured grids

A technique for constructing monotone, high resolution, multi‐dimensional upwind fluctuation distribution schemes for the scalar advection equation is presented. The method combines the second‐order Lax–Wendroff scheme with the upwind positive streamwise invariant (PSI) scheme via a fluctuation redistribution step, which ensures monotonicity (and which is a generalization of the flux‐corrected transport approach for fluctuation distribution schemes). Furthermore, the concept of a distribution point is introduced, which, when related to the equivalent equation for the scheme, leads to a ‘preferred direction’ for the limiting procedure, and hence to a new distribution of the fluctuation, which retains second‐order accuracy from the Lax–Wendroff scheme, even when the solution contains turning points. Experimental comparisons show that the new method compares favourably in terms of speed, accuracy and robustness with other, similar, techniques. Copyright © 2000 John Wiley & Sons, Ltd.     

A geometric mass‐preserving redistancing scheme for the level set function

In this paper we describe and evaluate a geometric mass‐preserving redistancing procedure for the level set function on general structured grids. The proposed algorithm is adapted from a recent finite element‐based method and preserves the mass by means of a localized mass correction. A salient feature of the scheme is the absence of adjustable parameters. The algorithm is tested in two and three spatial dimensions and compared with the widely used partial differential equation (PDE)‐based redistancing method using structured Cartesian grids. Through the use of quantitative error measures of interest in level set methods, we show that the overall performance of the proposed geometric procedure is better than PDE‐based reinitialization schemes, since it is more robust with comparable accuracy. We also show that the algorithm is well‐suited for the highly stretched curvilinear grids used in CFD simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Higher‐order upwind leapfrog methods for multi‐dimensional acoustic equations

A non‐dissipative and very accurate one‐dimensional upwind leapfrog method was successfully extended to higher‐order and multi‐dimensional acoustic equations. The governing equations in characteristic form and staggered grid were utilized to preserve the accuracy. Fourier analysis was performed to find the accurate scheme for acoustics and the resultant two‐dimensional methods were successfully applied to several classical test cases. Copyright © 2004 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.     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 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.     

Advances in the simulation of multi‐fluid flows with the particle finite element method. Application to bubble dynamics

In this work, we extend the Particle Finite Element Method (PFEM) to multi‐fluid flow problems with the aim of exploiting the fact that Lagrangian methods are specially well suited for tracking interfaces. We develop a numerical scheme able to deal with large jumps in the physical properties, included surface tension, and able to accurately represent all types of discontinuities in the flow variables. The scheme is based on decoupling the velocity and pressure variables through a pressure segregation method that takes into account the interface conditions. The interface is defined to be aligned with the moving mesh, so that it remains sharp along time, and pressure degrees of freedom are duplicated at the interface nodes to represent the discontinuity of this variable due to surface tension and variable viscosity. Furthermore, the mesh is refined in the vicinity of the interface to improve the accuracy and the efficiency of the computations. We apply the resulting scheme to the benchmark problem of a two‐dimensional bubble rising in a liquid column presented in Hysing et al. (International Journal for Numerical Methods in Fluids 2009; 60 : 1259–1288), and propose two breakup and coalescence problems to assess the ability of a multi‐fluid code to model topology changes. Copyright © 2010 John Wiley & Sons, Ltd.     

Adjoint‐based airfoil optimization with discretization error control

In this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.