Mathematical Problems in Engineering

Volume 2015, Article ID 142730, 11 pages

http://dx.doi.org/10.1155/2015/142730

## Explicit High Accuracy Maximum Resolution Dispersion Relation Preserving Schemes for Computational Aeroacoustics

^{1}The State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China^{2}Yichang Testing Technique Research Institute, No. 58 Shengli 3rd Road, Yichang, Hubei 430074, China^{3}Reactor Engineering Testing Research Center, China Nuclear Power Technology Research Institute Co., Ltd., Shenzhen 518026, China

Received 12 February 2015; Revised 21 May 2015; Accepted 26 May 2015

Academic Editor: Vassilios C. Loukopoulos

Copyright © 2015 J. L. Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A set of explicit finite difference schemes with large stencil was optimized to obtain maximum resolution characteristics for various spatial truncation orders. The effect of integral interval range of the objective function on the optimized schemes’ performance is discussed. An algorithm is developed for the automatic determination of this integral interval. Three types of objective functions in the optimization procedure are compared in detail, which show that Tam’s objective function gets the best resolution in explicit centered finite difference scheme. Actual performances of the proposed optimized schemes are demonstrated by numerical simulation of three CAA benchmark problems. The effective accuracy, strengths, and weakness of these proposed schemes are then discussed. At the end, general conclusion on how to choose optimization objective function and optimization ranges is drawn. The results provide clear understanding of the relative effective accuracy of the various truncation orders, especially the trade-off when using large stencil with a relatively high truncation order.

#### 1. Introduction

In a wide range of industrial fields such as aircraft, automobile, and turbomachinery flow induced noise and noise often interact with the turbulent flow strongly. These aeroacoustic problems usually involve sound waves of many octave bands, due to the large disparity in the characteristic scales of the acoustic and the flow fluctuations; high-order accurate methods with minimal dissipation and dispersion errors are required.

A recent review of high-order method can be found in [1, 2]. Four different families of high-order methods can be encountered in the literature: spectral and pseudospectral [3], Discontinuous Galerkin (DG) [4], Weighted Essentially Nonoscillatory (WENO) [5], and optimized finite difference (FD) methods [6–9].

Finite difference schemes are considered in the present work. Owing to the simplicity, a significant body of research has been devoted to optimize the finite difference method. Several finite difference schemes [6, 10–12] have been designed or optimized to resolve the waves with high accuracy and good spectral resolution. Lele [10] constructed such schemes by constraining some of the coefficients in the compact finite difference to achieve a certain truncation error and then determining the remainder of the coefficients by requiring the modified wave number to be equal to exact differentiation at certain wave numbers. Following a similar procedure, Tam and Webb [11] increased the resolution of a central finite difference approximation by minimizing the integrated dispersive (phase) errors in the wave number domain and proposed the dispersion-relation-preserving (DRP) scheme. The coefficients of the central discretization are determined by the truncation order and the minimization of errors. The fourth-order spatial central scheme of an optimized seven-point stencil shows better resolution characteristics than the standard maximum-order central finite difference schemes. However, as Zingg [13] pointed out, if the schemes are optimized too aggressively, they perform poorly for longer distances of travel. Furthermore, if a waveform has significant low wave number content, as in the case of a Gaussian pulse, the benefits of an optimized scheme can be minimal, and maximum-order schemes can even be superior. Cunha and Redonnet [14] also expressed a similar opinion on comparison of the optimization scheme and standard maximum-order scheme.

For the optimization of the finite difference scheme, there are two key factors that can affect the optimization result mostly. The first one is the optimization ranges for the integral modified wave number error. In Tam and Webb’s [11] first paper, this integral range is arbitrarily determined to . As Lockard et al. [15] pointed out, this was an aggressive choice. Later, in another paper by Tam et al. [8], they gave a more reasonable optimization range. However, they did not explain these values in detail. In Kim and Lee’s [16] paper on optimized compact scheme, this optimization range is determined by a trial and error method. In Cheong and Lee’s [17] paper on grid-optimized DRP scheme on General Geometries for Computational Aeroacoustics they just simply follow Tam’s integral ranges. More recently, Liu et al. [18] develop an algorithm to determine this range, but it is a little complex and did not give intuitive explanation; in this work, we will give a simpler algorithm for the optimization ranges; intuitive explanation is followed in discussion part. This simple algorithm can be readily adopted for optimized implicit compact scheme, nonuniform grid DRP scheme, upwind optimized DRP scheme, and curvilinear version of the above scheme too.

The second one is the truncation error order. Tam and Webb [11] choose the 4-order formal accuracy order for 7-point central scheme and larger stencil finite difference scheme too. However, as Zingg [13] and Cunha and Redonnet [14] pointed out, this scheme was not a good scheme for long distance travel wave; it is not better than the maximum-order schemes if there are many low frequency contents. So using a larger stencil scheme combined with higher truncation error order will be a natural compromised way. Kim and Lee [16] consider various truncation error orders for compact finite schemes. The similar various truncation error orders for explicit central finite difference schemes will be considered in this work.

The organization of the paper is as follows. In Section 2, brief introduction to spatial discretization of the central finite difference schemes is recalled; phase velocity and group velocity errors of the finite difference schemes in the wave number domain are given via Fourier analysis in Sections 3 and 4. Section 5 defines three types of minimum integral error objective functions for optimization. Section 6 describes the optimization strategies and shows a detailed procedure of optimization of the various truncated error order schemes. An intuitive simple algorithm was proposed in Section 7 to achieve a reasonable integral range. Additional discussion on optimization objective function is given in Section 7 as well. Then, the effectiveness of the various error orders optimized schemes is applied to one- and two-dimensional problems in Section 8. The conclusions are made in the last section.

#### 2. Spatial Discretization

Consider a uniform mesh size and the values of a function at nodes . The first-order spatial derivative of the function at can be approximated by a central, -point stencil, finite difference scheme aswhere the coefficients are such as and , providing a scheme without dissipation. For standard schemes, the coefficients are fully determined from matching the Taylor series to order accuracy, so that the maximum order is reached.

The relations between the coefficients of are derived by matching the Taylor series coefficients of various truncation orders. These relations are as follows.

*Second Order**Fourth Order**Sixth Order**Eighth Order**Tenth Order**Twelfth Order*

For standard centered finite difference schemes, the coefficients are fully derived from the above (2)–(7). When , we get standard schemes using 7, 9, 11, and 13 points, which are referred to as Std7p6o, Std9p8o, Std11p10o, and Std13p12o in this work and are of orders 6, 8, 10, and 12th, respectively. The Taylor series conditions for these schemes are listed as following: Std7p6o: (2)–(4), . Std9p8o: (2)–(5), . Std11p10o: (2)–(6), . Std13p12o: (2)–(8), .

For convenience, their coefficients are listed in Tables 1–3 (the last column) of Appendix. These standard schemes have unique coefficients, and these are the highest order ones obtainable with these schemes.