Elsevier

Journal of Computational Physics

Volume 190, Issue 2, 20 September 2003, Pages 459-477
Journal of Computational Physics

Optimized prefactored compact schemes

https://doi.org/10.1016/S0021-9991(03)00293-6Get rights and content

Abstract

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and dissipation errors. In this paper we describe a strategy for developing high-order accurate prefactored compact schemes, requiring very small stencil support. These schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes. The prefactorization strategy splits the central implicit schemes into forward and backward biased operators. Using Fourier analysis, we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components. This ensures that when the forward and backward stencils are added, the original central compact scheme is recovered. To extend the resolution characteristic of the schemes, an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh. The resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth- and eighth-order compact schemes making them more suitable for high-resolution numerical simulations in gas dynamics and computational aeroacoustics. The efficiency, accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems.

Introduction

The field of computational aeroacoustics (CAA) has grown rapidly during the last decade due to a resurgence of interest in aeroacoustic phenomena driven by ever harsher legislation and increasing environmental awareness. CAA is concerned with the accurate numerical prediction of aerodynamically generated noise as well as its propagation and far-field characteristics. The inherently unsteady nature of aeroacoustic phenomena, the disparity in magnitude between mean and acoustic flow quantities, and the high frequencies often encountered place stringent demands on the numerics [1]. The trend therefore within the field of CAA has been to employ higher-order accurate numerical schemes that have in some manner been optimized for wave propagation to reduce the required number of grid points per wavelength while still ensuring tolerable levels of numerical error.

Both the finite-volume and finite-difference methodologies have been adapted for this purpose [2], [3], [4], [5]. However, as multidimensional finite-volume algorithms are generally more expensive in terms of numerical cost than finite-difference algorithms, the majority of CAA codes are based on the finite-difference methodology. Finite-difference schemes may be classified as either explicit or implicit. Explicit schemes employ large computational stencils for accuracy. By comparison, implicit (compact) schemes achieve high-order accuracy by solving for the spatial derivatives as independent variables at each grid point. For the same stencil width, compact schemes are known to exhibit significantly more resolution of the smaller scales than an equivalent explicit scheme. The drawback with these schemes is their implicit nature, which necessitates the solution of a linear system of equations to obtain the spatial derivative at any point. Generally, however, these systems are of a narrow-banded nature (usually tridiagonal) and may therefore be solved quite efficiently.

The focus of the present paper is a family of small-stencil compact schemes recently proposed by Hixon [6]. These schemes use a prefactorization method to reduce a non-dissipative central-difference stencil to two lower-order biased stencils which have easily solved reduced matrices. In [6] this approach was used to derive schemes of up to eighth-order accuracy requiring only three-point stencils. The advantages of these schemes over traditional compact schemes arise from their reduced stencil size and the independent nature of the resultant factored matrices. It is well known that a major difficulty in dealing with high-order finite-difference schemes is the formulation of stable stencils near boundaries. By reducing the stencil size of the compact schemes the prefactorization method reduces the required number of boundary stencils thereby simplifying boundary specification. As demonstrated in [6] the prefactorization also enhances robustness enabling the use of boundary stencils that would otherwise lead to unstable systems when used in conjunction with the unfactored scheme. The prefactored schemes also make boundary condition implementation much more straightforward than the standard schemes. As detailed in [6], this is a consequence of the L + U factorization associated with the new schemes, which allows boundary corrections (e.g., to the normal derivative [7], [8]) to be applied much more easily and efficiently than with an LU based scheme such as the Thomas algorithm.

In this work, we extend the factorization concept to a broader class of compact schemes using a more general derivation strategy. Rather than using the algebraic manipulations detailed in [6], we develop an approach that combines Fourier analysis with the notion of a numerical wavenumber. The advantage of this approach, alongside its more general nature, is that it ensures the application of an optimization strategy, designed to enhance the wave propagation characteristics of the schemes, is straightforward. Using this approach we derive two optimized fourth-order accurate prefactored compact schemes. The proposed schemes exhibit better wave propagation characteristics than the standard compact schemes, whilst retaining the advantages of the prefactored schemes.

The paper is organized as follows. In Section 2.1, a brief review of the compact differencing methodology is presented, along with a definition of the numerical wavenumber. In Section 2.2 the strategy for developing the prefactored schemes is described and a sample application is presented. The optimization of the prefactored schemes for wave propagation is presented in Section 2.3. Section 2.4 provides details of the boundary stencils for use with the interior schemes. The stability characteristics of the prefactored compact schemes in conjunction with the developed boundary closures are analysed in Section 3 through numerical applications and eigenvalue analysis. In Section 4 the favourable properties of the derived schemes are demonstrated through their application to several benchmark problems.

Section snippets

Compact discretization

Following Lele [9], a general compact approximation to the first spatial derivative (∂f/∂x) may be written in the form:βDi−2+αDi−1+Di+αDi+1+βDi+2=cfi+3−fi−36Δx+bfi+2−fi−24Δx+afi+1−fi−12Δx,where Di is the spatial derivative of the function f. The relations between the coefficients a, b, c, α and β are derived by matching the Taylor series coefficients of various orders. The first unmatched coefficient determines the formal truncation error of the approximation. These relations are:a+b+c=1+2α+2β(

Scheme analysis

The computational cost of the optimized prefactored compact schemes is an important consideration. The optimization procedure itself does not incur any additional computational costs. The optimized schemes therefore require the same computational effort as the un-optimized schemes. The prefactorization does, however, incur a small computational penalty. To see this we first note that either of the optimized prefactored fourth-order schemes may be written in the form:12DjF=1FΔxbF(fj+1−fj)+dF(f

Applications

In this section we illustrate the properties of the derived schemes through their application to several benchmark problems. These problems are taken from the First [14] and Second [15] Workshops on Benchmark problems for Computational Acoustics.

Conclusions

A strategy for developing prefactored compact schemes has been presented. The approach facilitates the optimization of the biased stencils for the simulation of wave phenomena. The procedure employs Fourier analysis and the concept of a numerical wavenumber to determine the coefficients of the biased stencils. The optimized prefactored compact schemes have smaller stencil sizes and require only the solution of two independent bi-diagonal matrices. Third-order accurate boundary stencils have

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    Citation Excerpt :

    To overcome the disadvantage, finite difference schemes with better or spectral-like resolutions have been developed and widely investigated. The compact scheme [6–13,32] is an influential example. Lele [6] performed extensive analysis on compact schemes and showed that through the implicit relation between the derivatives and function values on several grid points, the error in large wavenumber region is reduced and resolutions for short waves are improved.

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