Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

Accurate computations of two‐dimensional turbulent hypersonic shock–shock interactions that arise when single and dual shocks impinge on the bow shock in front of a cylinder are presented. The simulation methods used are a class of lower–upper symmetric‐Gauss–Seidel implicit anti‐diffusive weighted essentially non‐oscillatory (WENO) schemes for solving the compressible Navier–Stokes equations with Spalart–Allmaras one‐equation turbulence model. A numerical flux of WENO scheme with anti‐diffusive flux correction is adopted, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of first‐order dissipative methods. Experimental flow fields of type IV shock–shock interactions with single and dual incident shocks by Wieting are computed. By using the WENO scheme with anti‐diffusive flux corrections, the present solution indicates that good accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Computed surface pressure distribution and heat transfer rate are also compared with experimental data and other computational results and good agreement is found. Copyright © 2009 John Wiley & Sons, Ltd.     

Linear stability of WENO schemes coupled with explicit Runge–Kutta schemes

This paper focuses on the results of the linear stability analysis of the finite‐difference weighted essentially non‐oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order‐optimised WENO schemes of the sixth and eighth order and the bandwidth‐optimised WENO schemes of the third and fourth order are considered. Several explicit Runge–Kutta schemes including the recently published strong stability‐preserving explicit Runge–Kutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the Courant–Friedrichs–Lewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high‐order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order accurate numerical solutions of incompressible flows with the artificial compressibility method

A high‐order accurate, finite‐difference method for the numerical solution of incompressible flows is presented. This method is based on the artificial compressibility formulation of the incompressible Navier–Stokes equations. Fourth‐ or sixth‐order accurate discretizations of the metric terms and the convective fluxes are obtained using compact, centred schemes. The viscous terms are also discretized using fourth‐order accurate, centred finite differences. Implicit time marching is performed for both steady‐state and time‐accurate numerical solutions. High‐order, spectral‐type, low‐pass, compact filters are used to regularize the numerical solution and remove spurious modes arising from unresolved scales, non‐linearities, and inaccuracies in the application of boundary conditions. The accuracy and efficiency of the proposed method is demonstrated for test problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

The blood flow model maintains the steady‐state solutions, in which the flux gradients are non‐zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted essentially non‐oscillatory (WENO) schemes to this model with such well‐balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well‐balanced property, and keep good resolution for smooth and discontinuous solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Positivity preserving pointwise implicit schemes with application to turbulent compressible flat plate flow

A family of positivity preserving pointwise implicit schemes applicable to source term dominated problems is constructed, where the minimum order of spatial accuracy is one and the maximum is three. It is designed for achieving steady state numerical solutions and is constructed through the analysis of appropriate model problems, where the convective fluxes for the higher‐order members are prescribed by the Chakravarthy–Osher family of total variation diminishing (TVD) schemes. Multidimensionality is facilitated by operator splitting. Numerical experimentation confirms the stability, convergence, accuracy, positivity, and computational efficiency associated with the proposed schemes. These schemes are ideally suited to solving the low‐Reynolds number turbulent k–ϵ equations for which the positivity of k and ϵ and the presence of stiff source terms are critical issues. Hence, using a finite volume formulation of these schemes, the low‐Reynolds number Chien k–ϵ turbulence model is implemented for a flat plate geometry and a series of turbulent flow (steady state) computations are carried out to demonstrate the positivity, robustness, and reliability of the algorithm. The free‐stream and initial k and ϵ values are specified in a very simple manner. Algorithm convergence acceleration is achieved using Multigrid techniques. The k–ϵ model flow predictions are shown to be in agreement with empirical profiles. Copyright © 2001 John Wiley & Sons, Ltd.     

A hybrid kinetic WENO scheme for inviscid and viscous flows

In this paper, we propose a high‐order finite volume hybrid kinetic Weighted Essentially Non‐Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high‐order accuracy and good shock‐capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one‐dimensional and two‐dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Third-order modified coefficient scheme based on essentially non-oscillatory scheme

A third-order numerical scheme is presented to give approximate solutions to multi-dimensional hyperbolic conservation laws only using modified coefficients of an essentially non-oscillatory （MCENO） scheme without increasing the base points during construction of the scheme. The construction process shows that the modified coefficient approach preserves favourable properties inherent in the original essentially nonoscillatory （ENO） scheme for its essential non-oscillation, total variation bounded （TVB）, etc. The new scheme improves accuracy by one order compared to the original one. The proposed MCENO scheme is applied to simulate two-dimensional Rayleigh-Taylor （RT） instability with densities 1：3 and 1：100, and solve the Lax shock-wave tube numerically. The ratio of CPU time used to implement MCENO, the .third-order ENO and fifth-order weighed ENO （WENO） schemes is 0.62：1：2.19. This indicates that MCENO improves accuracy in smooth regions and has higher accuracy and better efficiency compared to the original ENO scheme.     

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities,II

Problems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.     

Assessment of WENO schemes for multi‐dimensional Euler equations using GPU

This paper presents a detailed study on the implementation of Weighted Essentially Non‐Oscillatory (WENO) schemes on GPU. GPU implementation of up to ninth‐order accurate WENO schemes for the multi‐dimensional Euler equations of gas dynamics is presented. The implementation detail is discussed in the paper. The computational times of different schemes are obtained and the speedups are reported for different number of grid points. Furthermore, the execution times for the main kernels of the code are given and compared with each other. The numerical experiments show the speedups for the WENO schemes are very promising especially for fine grids. Copyright © 2014 John Wiley & Sons, Ltd.     

Improved third‐order weighted essentially nonoscillatory scheme

A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.     

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 low-dissipation third-order weighted essentially nonoscillatory scheme with a new reference smoothness indicator

The classical third-order weighted essentially nonoscillatory (WENO) scheme is notoriously dissipative as it loses the optimal order of accuracy at critical points and its two-point finite difference in the smoothness indicators is unable to differentiate the critical point from the discontinuity. In recent years, modifications to the smoothness indicators and weights of the classical third-order WENO scheme have been reported to reduce numerical dissipation. This article presents a new reference smoothness indicator for constructing a low-dissipation third-order WENO scheme. The new reference smoothness indicator is a nonlinear combination of the local and global stencil smoothness indicators. The resulting WENO-Rp3 scheme with the power parameter p=1.5 achieves third-order accuracy in smooth regions including critical points and has low dissipation, but numerical results show this scheme cannot keep the ENO property near discontinuities. The recommended WENO-R3 scheme (p=1) keeps the ENO property and performs better than several recently developed third-order WENO schemes.     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Flux‐difference splitting‐based upwind compact schemes for the incompressible Navier–Stokes equations

Third‐order and fifth‐order upwind compact finite difference schemes based on flux‐difference splitting are proposed for solving the incompressible Navier–Stokes equations in conjunction with the artificial compressibility (AC) method. Since the governing equations in the AC method are hyperbolic, flux‐difference splitting (FDS) originally developed for the compressible Euler equations can be used. In the present upwind compact schemes, the split derivatives for the convective terms at grid points are linked to the differences of split fluxes between neighboring grid points, and these differences are computed by using FDS. The viscous terms are approximated with a sixth‐order central compact scheme. Comparisons with 2D benchmark solutions demonstrate that the present compact schemes are simple, efficient, and high‐order accurate. Copyright © 2008 John Wiley & Sons, Ltd.     

Influence of spatial discretization and unsteadiness on the simulation of rocket combustors

This paper investigates some important numerical aspects for the simulation of model rocket combustors. Precisely, (1) a new high‐order discretization technique (multi‐dimensional limiting process (MLP), low diffusion, and MLP^ld) is presented and compared with conventional second‐order schemes with different flux limiters. (2) Time accurate unsteady Reynolds‐averaged Navier–Stokes (RANS) simulations are performed to assess possible improvements in comparison with steady‐state RANS simulations. (3) Fully 3D simulations of an axisymmetric rocket combustor are compared with 2D axisymmetric ones. All studies are based on the Penn State preburner combustor experiment, which uses gaseous oxygen and hydrogen. This comprehensive study offers unique insight into how the mentioned numerical influence factors change the flow field, flame, and wall heat fluxes in the model rocket combustor. Because wall heat fluxes are known from the experiment only, numerical results are compared with LES of other authors, too. It will be shown that the high‐order spatial discretization significantly improves the agreement with measured wall heat fluxes at low additional computational cost. In general the transition from simple to more complex numerical approaches steadily improves the qualitative agreement between simulation and experiment. Copyright © 2015 John Wiley & Sons, Ltd.