Central Compact Scheme Research Articles

Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation

The dispersion analysis of numerical schemes for finite difference computations based on 2D compressible linearized Navier–Stokes equation (LNSE) is presented here. The analysis presents a more holistic view of the applicability of the schemes while solving compressible NSE, rather than analyzing model 1D or 2D convection or convection–diffusion equations. A theoretical dispersion analysis of model 2D−LNSE is performed first which categorizes it into vortical, entropic and two acoustic modes. The variation of the theoretical dispersion relation for each of the modes with wavenumber, Mach number and Reynolds number is analyzed. It is noted that the entropic mode is the most diffusive among all the modes. Two acoustic modes are less diffusive than the vortical mode up to a certain absolute wavenumber Kb depending upon the Mach number and Reynolds number. It is also shown that while vortical and entropic modes are non-dispersive in nature, the two acoustic modes are dispersive only up to K=Kb. Next, numerical dispersion and diffusion corresponding to each of the modes are analyzed for second order central difference (CD2) and sixth order central compact difference scheme proposed in Lele (J. Comput. Phys., Vol-103(1), 1992) for discretization of convective and diffusive derivatives using fourth-order Runge–Kutta (RK4) time integration method. Results show the existence of numerical unstable zone and regions where spurious upstream propagating waves are noted for each of the modes in the spectral plane. From these and the relative size of the dispersion relation preserving zones for the modes, applicability of the numerical schemes can be assessed. Analysis of the variation of the numerical unstable zone with Mach number, orientation of the mean flow and the unit CFL number Nc=Δt/Δx for each of the modes are presented. It is noted that entropic mode displays the lowest critical CFL number of all the four modes. Lastly, linearized and nonlinear numerical simulations for the convection of acoustic and vortical pulse with varying spectral content are presented. The linearized simulations are also compared with corresponding exact solutions. Obtained results indicate a direct correspondence of the stability analysis performed based on 2D−LNSE even for nonlinear simulations which result in additional intra modal energy transfer and interactions.

The optimization of the compact upwind scheme for incompressible flow

Different compact upwind schemes have been developed and used to numerically approximate a convection term in the Navier–Stokes equation. With different point stencils, the compact upwind schemes are mainly classified as the central, the function-biased, and the derivative-biased compact upwind schemes. They have different numerical characteristics. In this paper, by using Fourier analysis and numerical test, it is found that the function-biased compact upwind schemes have better resolution properties than the derivative-biased compact upwind schemes. Furthermore, an optimization method named dispersion-dissipation-balancing (DDB) optimization is proposed, by which better spectral resolution of these schemes is obtained by optimizing coefficients of these schemes based on the balance between a dispersion error and a dissipation error. Compared with the popular dispersion-relation-preserving (DRP) optimization method, the schemes optimized by the DDB method have proper dispersion and dissipation errors. They eliminate both the nonphysical oscillations and spurious vortices in the numerical case of the double shear layers flow. In addition, the central compact upwind scheme optimized by the DDB method (OCCUS_DDB) has the best performance among the schemes studied in this paper.

Two Dimensional Analysis of Hybrid Spectral/Finite Difference Schemes for Linearized Compressible Navier\u2013Stokes Equations

This study seeks to compare different combinations of spatial dicretization methods under a coupled spatial temporal framework in two dimensional wavenumber space. The aim is to understand the effect of dispersion and dissipation on both the convection and diffusion terms found in the two dimensional linearized compressible Navier–Stokes Equations (LCNSE) when a hybrid finite difference/Fourier spectral scheme is used in the x and y directions. In two dimensional wavespace, the spectral resolution becomes a function of both the wavenumber and the wave propagation angle, the orientation of the wave front with respect to the grid. At sufficiently low CFL number where temporal discretization effects can be neglected, we show that a hybrid finite difference/Fourier spectral schemes is more accurate than a full finite difference method for the two dimensional advection equation, but that this is not so in the case of the LCNSE. Group velocities, phase velocities as well as numerical amplification factor were used to quantify the numerical anisotropy of the dispersion and dissipation properties. Unlike the advection equation, the dispersion relation representing the acoustic modes of the LCNSE contains an acoustic terms in addition to its advection and viscous terms. This makes the group velocity in each spatial direction a function of the wavenumber in both spatial directions. This can lead to conditions for which a hybrid Fourier spectral/finite difference method can become less or more accurate than a full finite difference method. To better understand the comparison of the dispersion properties between a hybrid and full FD scheme, the integrated sum of the error between the numerical group velocity V^{*}_{grp,full} and the exact solution across all wavenumbers for a range of wave propagation angle is examined. In the comparison between a hybrid and full FD discretization schemes, the fourth order central (CDS4), fourth order dispersion relation preserving (DRP4) and sixth order central compact (CCOM6) schemes share the same characteristics. At low wave propagation angle, the integrated errors of the full FD and hybrid discretization schemes remain the same. At intermediate wave propagation angle, the integrated error of the full FD schemes become smaller than that of the hybrid scheme. At large wave propagation angle, the integrated error of the full FD schemes diverges while the integrated error of the hybrid discretization schemes converge to zero. At high reduced wavenumber and sufficiently low CFL number where temporal discretization error can be neglected, it was found that the numerical dissipation of the viscous term based on the CDS4, DRP4, CCOM6 and isotropy optimized CDS4 schemes (hbox {CDS4}_{{opt}}) schemes was lower than the actual physical dissipation, which is only a function of the cell Reynolds number. The wave propagation angle at which the numerical dissipation of the viscous term approaches its maximum occurs at pi /4 for the CDS4, DRP4, CCOM6 and hbox {CDS4}_{{opt}} schemes.

A High Accuracy Preserving Parallel Algorithm for Compact Schemes for DNS

A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.

Effects of compressibility and Atwood number on the single-mode Rayleigh-Taylor instability

In order to study the effect of compressibility on Rayleigh-Taylor (RT) instability, we numerically simulated the late-time evolution of two-dimensional single-mode RT instability for isothermal background stratification with different isothermal Mach numbers and Atwood numbers (At) using a high-order central compact finite difference scheme. It is found that the initial density stratification caused by compressibility plays a stabilizing role, while the expansion-compression effect of flow plays a destabilizing role. For the case of small Atwood number, the density difference between the two sides of the interface is small, and the density distribution of the upper and lower layers is nearly symmetrical. The initial density stratification plays a dominant role, and the expansion-compression effect has little influence. With the increase in the Atwood number, the stabilization effect of initial density stratification decreases, and the instability caused by the expansion-compression effect becomes more significant. The flow structures of bubbles and spikes are quite different at medium Atwood number. The effect of compressibility on the bubble velocity is strong at large At. The bubble height is approximately a quadratic function of time at potential flow growth stage. The average bubble acceleration is nearly proportional to the square of Mach number at At = 0.9.

Solution of Euler equations for unsteady flow problems using a central compact scheme

We solve the Euler equations in one and two dimensions using a central compact scheme coupled with an AUSM+ algorithm. The compact scheme is used in the form of a cell-face interpolation scheme to create the left- and right-states of the primitive variables required by the AUSM+ based solver. We invoke a TVD limiter to locally apply smaller stencil low-order dissipative formulae near shocks. No extra high-order artificial dissipation is added to stabilize the computation. A number of one and two dimensional test cases has been solved to show the effectiveness of this approach.

High Resolution Compact Finite Difference Schemes for Convection Dominated Problems

In this short article, the upwind and central compact finite difference schemes for spatial discretization of the first-order derivative are analyzed. Comparison of the schemes is provided and the best discretization scheme for convection dominated problems is suggested.

A new central compact finite difference scheme with high spectral resolution for acoustic wave equation

Based on the existing cell-node and cell-centered compact finite difference schemes, we developed a new central compact scheme with a high spectral resolution for the acoustic wave equation. In the new scheme, both the function values on the cell-nodes and cell-centers are used to compute the second-order spatial derivatives on the cell-nodes. The cell-centered values are stored and updated as independent variables in the modeling. The spatial derivatives on the cell-centers are evaluated by half shifting the indices in the formula designed for the cell-nodes. Compared to the conventional compact interpolation scheme, the proposed approach can avoid introducing transfer errors. Either Taylor-series expansion-based or optimized least-squares-based methods are used to calculate the finite difference coefficients. Theoretical analysis and synthetic examples demonstrate that the optimized least-squares-based method can provide higher accuracy than the Taylor-series expansion-based method. This new scheme is not a simple combination of the cell-node and cell-centered compact schemes and outperforms them in three scenarios. Firstly, it can promise higher accuracy considering the same formal truncation errors and model parameters. Thus, it can maintain superior precision while using a shorter spatial finite difference stencil. Secondly, compared to the cell-node compact scheme with half of grid spacing, the new scheme can yield equally as accurate results with less time consuming, together with saving approximately 25% and 29% of memory in 2D and 3D modeling, respectively. Finally, for similar memory requirements, the new method can more efficiently provide solutions with higher accuracy. The synthetic examples on the 2D homogeneous and the 3D horizontally-layered models demonstrate the advantages of the proposed scheme. The numerical simulations with 2D Marmousi model further validate its accuracy, efficiency and flexibility in complex media.

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACTThis paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.

An implementation of the direct-forcing immersed boundary method using GPU power

ABSTRACTA graphics processing unit (GPU) is utilized to apply the direct-forcing immersed boundary method. The code running on the GPU is generated with the help of the Compute Unified Device Architecture (CUDA). The first and second spatial derivatives of the incompressible Navier-Stokes equations are discretized by the sixth-order central compact finite-difference schemes. Two flow fields are simulated. The first test case is the simulated flow around a square cylinder, with the results providing good estimations of the wake region mechanics and vortex shedding. The second test case is the simulated flow around a circular cylinder. This case was selected to better understand the effects of sharp corners on the force coefficients. It was observed that the estimation of the force coefficients did not result in any troubles in the case of a circular cylinder. Additionally, the performance values obtained for the calculation time for the solution of the Poisson equation are compared with the values for other CPUs and GPUs from the literature. Consequently, approximately 3× and 20× speedups are achieved in comparison with GPU (using CUSP library) and CPU, respectively. CUSP is an open-source library for sparse linear algebra and graph computations on CUDA.

The standard upwind compact difference schemes for incompressible flow simulations

Compact difference schemes have been used extensively for solving the incompressible Navier–Stokes equations. However, the earlier formulations of the schemes are of central type (called central compact schemes, CCS), which are dispersive and susceptible to numerical instability. To enhance stability of CCS, the optimal upwind compact schemes (OUCS) are developed recently by adding high order dissipative terms to CCS. In this paper, it is found that OUCS are essentially not of the upwind type because they do not use upwind-biased but central type of stencils. Furthermore, OUCS are not the most optimal since orders of accuracy of OUCS are at least one order lower than the maximum achievable orders. New upwind compact schemes (called standard upwind compact schemes, SUCS) are developed in this paper. In contrast to OUCS, SUCS are constructed based completely on upwind-biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations. Furthermore, SUCS can achieve the maximum achievable orders of accuracy and hence be more compact than OUCS. More importantly, SUCS have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments.

Development of high-order weighted compact schemes with various difference methods

This paper is concerned with the development of high-order weighted compact schemes based on linear central compact schemes (Liu et al., 2013). For the purpose of capturing strong discontinuities, the high-order upwind weighted nonlinear interpolations are combined with the central compact schemes and cell-centered compact schemes to achieve two different weighted compact schemes, called the CCSSR-W and CCCSSR-W schemes respectively. The CCSSR-W schemes are developed up to ninth order, and four different nonlinear weights of CCSSR-W schemes are investigated in detail. As an improvement of the CCSSR-HW scheme (Liu et al., 2015), the hybrid weighted nonlinear interpolation is substituted into the cell-centered compact scheme to obtain the CCCSSR-HW scheme. For all the proposed schemes, the flux vector splitting method is carried out by coupling with characteristic projections on the conservative fluxes. Through the systematic accuracy tests, spectral analysis and numerical experiments, the results demonstrate that the higher order CCSSR-W scheme is more accurate and effective than the lower order ones. The CCCSSR-W scheme has better accuracy and resolution than the original WENO scheme. The CCCSSR-HW scheme has higher resolution and lower dissipation than the CCSSR-HW scheme, and even better than the higher order CCSSR-W schemes.

Numerical dissipation control in high order shock-capturing schemes for LES of low speed flows

The Yee & Sjögreen adaptive numerical dissipation control in high order scheme (High Order Filter Methods for Wide Range of Compressible Flow Speeds, ICOSAHOM 09, 2009) is further improved for DNS and LES of shock-free turbulence and low speed turbulence with shocklets. There are vastly different requirements in the minimization of numerical dissipation for accurate turbulence simulations of different compressible flow types and flow speeds. Traditionally, the method of choice for shock-free turbulence and low speed turbulence are by spectral, high order central or high order compact schemes with high order linear filters. With a proper control of a local flow sensor, appropriate amount of numerical dissipation in high order shock-capturing schemes can have spectral-like accuracy for compressible low speed turbulent flows. The development of the method includes an adaptive flow sensor with automatic selection on the amount of numerical dissipation needed at each flow location for more accurate DNS and LES simulations with less tuning of parameters for flows with a wide range of flow speed regime during the time-accurate evolution, e.g., time varying random forcing. An automatic selection of the different flow sensors catered to the different flow types is constructed. A Mach curve and high-frequency oscillation indicators are used to reduce the tuning of parameters in controlling the amount of shock-capturing numerical dissipation to be employed for shock-free turbulence, low speed turbulence and turbulence with strong shocks. In Kotov et al. (High Order Numerical Methods for LES of Turbulent Flows with Shocks, ICCFD8, Chengdu, Sichuan, China, July 14–18, 2014) the LES of a turbulent flow with a strong shock by the Yee & Sjögreen scheme indicated a good agreement with the filtered DNS data. A work in progress for the application of the adaptive flow sensor for compressible turbulence with time-varying random forcing is forthcoming. The present study examines the versatility of the Yee & Sjögreen scheme for DNS and LES of traditional low speed flows without forcing. Special attention is focused on the accuracy performance of this scheme using the Smagorinsky and the Germano–Lilly SGS models.

A new central compact finite difference formula for improving robustness in weighted compact nonlinear schemes

The weighted compact nonlinear scheme (WCNS) is a typical well-known finite difference method that shows high-order accuracy and high resolution when applied to problems involving hyperbolic conservation laws. However, when applied to flow problems with severe discontinuities, the original WCNS often suffers from negative densities or negative pressures. We have found that the robustness of the WCNS can be remarkably improved by changing the original central finite difference scheme employed in the WCNS to an alternative one. In this article, a new cell-edge- and cell-node-type alternate central compact scheme (ACCS) is proposed, and a new WCNS (i.e., alternate central weighted compact nonlinear scheme (ACWCNS)) is developed by combining the ACCS with a nonlinear reconstruction (weighted interpolation) method. To elucidate the basic characteristics of the ACWCNS, we perform truncation error analysis, wavenumber analysis, semi-discrete eigenvalue analysis, and accuracy validation for 1D advection equations. Subsequently, the ACWCNS is applied to several 1D and 2D benchmark problems of compressible flows with severe discontinuities. When solving such problems, it is difficult to preserve the positivity of both density and pressure using the conventional WCNS; nevertheless, the ACWCNS is able to provide stable solutions. In the smooth regions of the solutions, the spatial resolution of the scheme is found to deteriorate slightly as the robustness of the method improved. To compensate for this, the nonlinear reconstruction method in the ACWCNS is modified and the obtained computational results are compared with those obtained by alternative approaches.

High accuracy solution of bi-directional wave propagation in continuum mechanics

Solution of partial differential equations by numerical method is strongly affected due to numerical errors, which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation. To quantify and control such errors and obtain high accuracy solutions, we consider a class of problems which involve second derivative of unknowns with respect to time. Here, we analyse numerical metrics such as the numerical group velocity, numerical phase speed and the numerical amplification factor for different methods in solving the model bi-directional wave equation (BDWE). Such equations can be solved directly, for example, by Runge–Kutta–Nyström (RKN) method. Alternatively, the governing equation can be converted to a set of first order in time equations and then using four-stage fourth order Runge–Kutta (RK4) method for time integration. Spatial discretisation considered are the classical second and fourth order central difference schemes, along with Lele's central compact scheme for evaluating second derivatives. In another version, we have used Lele's scheme for evaluating first derivatives twice to obtain the second derivative. As BDWE represents non-dissipative, non-dispersive dynamics, we also consider the canonical problem of linearised rotating shallow water equation (LRSWE) in a new formulation involving second order derivative in time, which represents dispersive waves along with a stationary mode. The computations of LRSWE with RK4 and RKN methods for temporal discretisation and Lele's compact schemes for spatial discretisation are compared with computations performed with RK4 method for time discretisation and staggered compact scheme (SCS) for spatial discretisation by treating it as a set of three equations as reported in Rajpoot et al. (2012) [1].

A central compact scheme for numerical solution of two-phase incompressible flow using Allen–Cahn phase field model

This article presents a numerical method for the Allen–Cahn phase field model coupled with the incompressible Navier–Stokes equations for simulating two-phase incompressible flow. The numerical method is based on the artificial compressibility method with dual-time stepping technique applied to the governing equations written in conservative form. The convective terms are approximated by using an eighth-order central compact scheme with filtering while the viscous terms are approximated by a sixth-order central compact scheme. Approximate factorization-based alternating direction implicit algorithm is used to solve the discretized linear system. The effectiveness of the method is demonstrated by computing several benchmark two-phase incompressible flow problems.

Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves

In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two-step method. In the first step, the sixth-order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first-order derivative term. For the purpose of retaining both of the long-term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure-like variable is approximated by the sixth-order accurate three-point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high-order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first-order advection term κux in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015

Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes

In this paper, we generalize the so-called inverse Lax−Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144–8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235–256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundstrom, Math. Comput. 26 (1972) 649–686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge−Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.

A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes

In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution (X. Liu et al., 2013 [20]). In our approach, we compute the flux derivatives on the cell-nodes by the physical fluxes on the cell nodes and numerical fluxes on the cell centers. To acquire the numerical fluxes on the cell centers, we perform a weighted hybrid interpolation of an upwind interpolation and a central interpolation. Through systematic analysis and numerical tests, we show that our nonlinear compact scheme has high order, high resolution and low dissipation, and has the same ability to capture strong discontinuities as regular weighted essentially non-oscillatory (WENO) schemes. It is a good choice for the simulation of multiscale problems with shock waves.

Developing Hybrid cell-edge and cell-node Dissipative Compact Scheme for Complex Geometry Flows

Developing high resolution finite difference scheme and enabling the use of this scheme on complex geometry are the aims of this study. High resolution has been achieved by Dissipative Compact Schemes (DCS), however, according to the recent research, applications of DCS on complex geometry may have serious problem for that the Geometric Conservation Law (GCL) is not satisfied, and this may cause numerical instability. To cope with this problem, a new scheme named Hybrid cell-edge and cell-node Dissipative Compact Scheme (HDCS) has been formulated. The formulation of the HDCS contains two steps. First, a new central compact scheme is formulated for the purpose of conveniently fulfilling the GCL, and then dissipation is added on the central scheme by high-order dissipative interpolation of cell-edge variables. The solutions of Euler and Navier-Stokes equations show that the HDCS can be applied successfully on complex geometry, while the DCS may suffer numerical instabilities. Moreover, high resolution of the HDCS may be observed in the test of scattering of acoustic waves by multiple cylinders.