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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 783101, 30 pages
http://dx.doi.org/10.1155/2012/783101
Research Article

The Technique of MIEELDLD in Computational Aeroacoustics

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

Received 3 January 2012; Accepted 14 February 2012

Academic Editor: M. F. El-Amin

Copyright © 2012 A. R. Appadu. 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

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and low dissipation errors. A technique has recently been devised in a Computational Fluid Dynamics framework which enables optimal parameters to be chosen so as to better control the grade and balance of dispersion and dissipation in numerical schemes (Appadu and Dauhoo, 2011; Appadu, 2012a; Appadu, 2012b; Appadu, 2012c). This technique has been baptised as the Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) and has successfully been applied to numerical schemes discretising the 1-D, 2-D, and 3-D advection equations. In this paper, we extend the technique of MIEELDLD to the field of computational aeroacoustics and have been able to construct high-order methods with Low Dispersion and Low Dissipation properties which approximate the 1-D linear advection equation. Modifications to the spatial discretization schemes designed by Tam and Webb (1993), Lockard et al. (1995), Zingg et al. (1996), Zhuang and Chen (2002), and Bogey and Bailly (2004) have been obtained, and also a modification to the temporal scheme developed by Tam et al. (1993) has been obtained. These novel methods obtained using MIEELDLD have in general better dispersive properties as compared to the existing optimised methods.

1. Introduction

Computational aeroacoustics (CAA) has been given increased interest because of the need to better control noise levels from aircrafts, trains, and cars due to increased transport and stricter regulations from authorities [1]. Other applications of CAA are in the simulation of sound propagation in the atmosphere to the improved design of musical instruments.

In computational aeroacoustics, the accurate prediction of the generation of sound is demanding due to the requirement for preservation of the shape and frequency of wave propagation and generation. It is well known [2, 3] that, in order to conduct satisfactory computational aeroacoustics, numerical methods must generate the least possible dispersion and dissipation errors. In general, higher order schemes would be more suitable for CAA than the lower-order schemes since, overall, the former are less dissipative [4]. This is the reason why higher-order spatial discretisation schemes have gained considerable interest in computational aeroacoustics.

The field of CAA has grown rapidly during the last decade and there has been a resurgence of interest in aeroacoustic phenomena characterised by 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. CAA involves mainly the development of numerical methods which approximate derivatives in a way that better preserves the physics of wave propagation unlike typical aerodynamic computations [1].

Since 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 [5]. The trend within the field of CAA has been to employ higher-order accurate numerical schemes that have in some manner been optimized for wave propagation in order to reduce the number of grid points per wavelength while ensuring tolerable levels of numerical error. Apart from acoustics and aeroacoustics, low amplitude wave propagation takes place over distances characterized by large multiples of wavelength in other areas such as [6](1)electromagnetics, for microcircuit design,(2)elastodynamics, for nondestructive testing,(3)seismology, for oil exploration,(4)medical Imaging, for accurate diagnosis,(5)hyperthermia, for noninvasive surgery.

Aerodynamics and other areas of fluid mechanics have benefitted immensely from the development of CFD [7]. The numerical analysis of flows around full aircraft configurations has become feasible with advances in both numerical techniques and computing machines. The temptation to apply effective CFD methods to aeroacoustic problems has been unavoidable and has been met with some success, but, in some cases, it has been observed that there is a necessity for some numerical protocols specific to problems involving disturbance propagation over long distances. The difference between aerodynamic and aeroacoustic problems lies mainly in the fact that for aeroacoustic computations, the solution is desired at some large distance from the aerodynamic source, but, in the case of aerodynamic problems, flow properties are required accurately only on the body itself [7]. Most aerodynamics problems are time independent, whereas aeroacoustics problems are, by definition, time dependent [8]. There are computational issues that are unique to aeroacoustics. Thus, computational aeroacoustics requires somewhat independent thinking and development.

At specific Courant numbers and angles of propagation, the perfect-shift property can be obtained, leading to exact propagation for all wavenumbers [9]. The perfect shift property refers to the situation when the error from the spatial discretisation precisely cancels that from the temporal discretisation. Several numerical schemes which combine the spatial and temporal discretisation produce the perfect shift property at specific Courant numbers [9]. Often this perfect cancellation of temporal and spatial errors occurs at cfl = 1.0. For such methods, the sum of the spatial and temporal error increases when the cfl number is decreased as the temporal error no longer cancels the spatial error. As the cfl number tends to zero, so does the temporal error and thus only spatial error remains. For most schemes, a low cfl represents the worst case associated with large dispersion or large dissipation errors as there is no cancellation of temporal and spatial errors [9]. Thus it is important to assess numerical methods over a range of Courant numbers [9]. However, this is not an issue for schemes built up from a high-accuracy spatial discretisation with a high-accuracy time-marching method. These schemes generally do not rely on cancellation to achieve high accuracy and thus the error does not increase as the Courant number is reduced.

The imaginary part of the numerical wavenumber represents numerical dissipation only when it is negative [10]. Due to the difference between the physical and numerical wavenumber, some wavenumbers propagate faster or slower than the wave speed of the original partial differential equation [11]. This is how dispersion errors are induced. The real part of the modified wavenumber determines the dispersive error while the imaginary part determines the dissipative error [9]. The group velocity of a wavepacket governs the propagation of energy of the wavepacket. The group velocity is characterised by Re((𝑑/(𝑑(𝜃)))(𝜃))1.0 which must be almost one in order to reproduce exact result [12]. Otherwise, dispersive patterns appear. When Re((𝑑/(𝑑(𝜃)))(𝜃))=1.0, the scheme has the same group velocity or speed of sound as the original governing equations and the numerical waves are propagated with correct wave speeds.

2. Organisation of Paper

This paper is organised as follows. In Section 3, we briefly describe the technique of Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) when used to optimise parameters in numerical methods. We also describe how this technique can be extended to construct high order methods with low dispersive and low dissipative properties in computational aeroacoustics. In Sections 48, we use MIEELDLD to obtain some optimized spatial methods based on a modification of the methods constructed by Tam and Webb [3], Lockard et al. [13], Zingg et al. [14], Zhuang and Chen [15], Bogey and Bailly [16]. Section 9 introduces an optimised temporal scheme which is obtained using MIEELDLD and based on a modification of the temporal discretisation method constructed by Tam et al. [17]. In Section 10, we construct numerical methods based on blending each of the five new spatial schemes with the new time discretisation scheme when used to discretise the 1-D linear advection equation and obtain rough estimates of the range of stability of these methods. Section 12 highlights the salient features of the paper.

3. The Concept of Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation

In this section, we describe briefly the technique of Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD). This technique have been introduced in Appadu and Dauhoo [18] and Appadu and Dauhoo [19]. We now give a resume of how we have derived this technique of optimisation.

Suppose the amplification factor of the numerical scheme when applied to the 1-D linear advection equation:𝜕𝑢𝜕𝑡+𝛽𝜕𝑢𝜕𝑥=0(3.1) is𝜉=𝐴+𝐼𝐵.(3.2) Then the modulus of the Amplification Factor (AFM) and the relative phase error (RPE) are calculated as||𝜉||,1AFM=RPE=𝑟𝑤tan1𝐵𝐴,(3.3) where 𝑟 and 𝑤 are the cfl number and phase angle, respectively.

For a scheme to have Low Dispersion and Low Dissipation, we require||||1RPE+(1AFM)0.(3.4)

The quantity, |1RPE| measures dispersion error while (1AFM) measures dissipation error. Also when dissipation neutralises dispersion optimally, we have||||||||1RPE(1AFM)0.(3.5) Thus on combining these two conditions, we get the following condition necessary for dissipation to neutralise dispersion and for low dispersion and low dissipation character to be satisfied:||||||||+||||+eldld=1RPE(1AFM)1RPE(1AFM)0.(3.6)

Similarly, we expect||||||||||||+eeldld=exp1RPE(1AFM)+exp1RPE(1AFM)20,(3.7) in order for Low Dispersion and Low Dissipation properties to be achieved.

The measure, eeldld, denotes the exponential error for low dispersion and low dissipation. The reasons why we prefer eeldld over eldld is because the former is more sensitive to perturbations.

We next explain how the integration process is performed in order to obtain the optimal parameter(s).

Only One Parameter Involved
If the cfl number is the only parameter, we compute 𝑤10eeldld𝑑𝑤,(3.8) for a range of 𝑤[0,𝑤1], and this integral will be a function of 𝑟. The optimal cfl is the one at which the integral quantity is closest to zero.

Two Parameters Involved
We next consider a case where two parameters are involved and whereby we would like to optimise these two parameters.
Suppose we want to obtain an improved version of the Fromm’s scheme which is made up of a linear combination of Lax-Wendroff (LW) and Beam-Warming (BW) schemes. Suppose we apply BW and LW in the ratio 𝜆1𝜆. This can be done in two ways.
In the first case, if we wish to obtain the optimal value of 𝜆 at any cfl, then we compute 𝑟10𝑤10eeldld𝑑𝑤𝑑𝑟,(3.9) which will be in terms of 𝜆.
The value of 𝑟1 is chosen to suit the region of stability of the numerical scheme under consideration while 𝑤1 is chosen such that the approximated RPE0 for 𝑟[0,𝑟1]. In cases where phase wrapping phenomenon does not occur, we use the exact RPE instead of the approximated RPE and in that case, 𝑤[0,𝜋].
The second way to optimise a scheme made up of a linear combination of Beam-Warming and Lax-Wendroff is to compute the IEELDLD as 𝑤10eeldld𝑑𝑤 and the integral obtained in that case will be a function of 𝑟 and 𝜆. Then a 3-D plot of this integral with respect to 𝑟[0,𝑟1] and 𝜆[0,1] enables the respective optimal values of 𝑟 and 𝜆 to be located. The optimised scheme obtained will be defined in terms of both a cfl number and the optimal value of 𝜆 to be used.

Considerable and extensive work on the technique of Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation has been carried out in Appadu and Dauhoo [18], Appadu and Dauhoo [19], Appadu [2022].

In Appadu and Dauhoo [18], we have obtained the optimal cfl for some explicit methods like Lax-Wendroff, Beam-Warming, Crowley, Upwind Leap-Frog, and Fromm’s schemes. In Appadu and Dauhoo [19], we have combined some spatial discretisation schemes with the optimised time discretisation method proposed by Tam and Webb [3] in order to approximate the linear 1-D advection equation. These spatial derivatives are a standard 7-point and 6th-order central difference scheme (ST7), a standard 9-point and 8th-order central difference scheme (ST9) and optimised spatial schemes designed by Tam and Webb [3], Lockard et al. [13], Zingg et al. [14], Zhuang and Chen [15] and Bogey and Bailly [16]. The results from some numerical experiments were quantified into dispersion and dissipation errors and we have found that the quality of the results is dependent on the choice of the cfl number even for optimised methods, though to a much lesser degree as compared to standard methods.

Moreover, in Appadu [20], we obtain the optimal cfl of some multilevel schemes in 1-D. These schemes are of high order in space and time and have been designed by Wang and Liu [23]. We have also optimised the parameters in the family of third-order schemes proposed by Takacs [24]. The optimal cfl of the 2-D CFLF4 scheme which is a composite method made up of Corrected Lax-Friedrichs and the two-step Lax-Friedrichs developed by Liska and Wendroff [25] has been computed and some numerical experiments have been performed such as 2-D solid body rotation test [26], 2-D acoustics [27], and 2-D circular Riemann problem [26]. We have shown that better results are obtained when the optimal parameters obtained using MIEELDLD are used.

Some more interesting features of MIEELDLD are detailed in Appadu [21]. In that paper, we extend the measures used by Tam and Webb [3], Bogey and Bailly [16], Berland et al. [28] in a computational aeroacoustics framework to suit them in a computational fluid dynamics framework such that the optimal cfl of some known numerical methods can be obtained. Thus, we define the following integrals: integrated error from Tam and Webb [3], (IETAM), integrated error from Bogey and Bailly [16] ((IEBOGEY), and integrated error from Berland et al. [28] (IEBERLAND) as follows:IETAM=𝑤10||||1RPE2𝑑𝑤,IEBOGEY=𝑤10||||1RPE𝑑𝑤,IEBERLAND=𝑤10||||(1𝛼)1RPE+𝛼(1AFM)𝑑𝑤.(3.10)

The optimal cfl is obtained by plotting the respective integral with respect to the cfl number and locating the cfl at which the integral is least. The techniques used to obtain the quantities IETAM, IEBOGEY, and IEBERLAND are named Minimised Integrated Error from Tam and Webb [3] (MIETAM), Minimised Integrated Error from Bogey and Bailly [16] (MIEBOGEY), and Minimised Integrated Error from Berland et al. [28] (MIEBERLAND) respectively. It is shown that MIEELDLD has an upper hand over the other techniques of optimisation: MIETAM, MIEBOGEY, and MIEBERLAND.

The work in Appadu [22] helps us to understand why not all composite schemes can be effective to capture shocks with minimum dispersion and dissipation. The findings concluded are that some efficient composite methods to approximate the 1-D linear advection equation are as follows: composite methods using Lax-Wendroff and Beam-Warming as either predictor or corrector steps, a linear combination of either Lax-Wendroff and Beam-Warming schemes or MacCormack and two-step Lax-Friedrichs and the composite MacCormack/Lax-Friedrichs schemes [29].

4. Modification to Space Discretisation Scheme Proposed by Tam and Webb [3]

Tam and Webb [3] constructed a 7-pt and 4th-order central difference method based on a minimization of the dispersion error.

They approximated 𝜕𝑢/𝜕𝑥 at 𝑥0 as𝜕𝑢=1𝜕𝑥3𝑖=3𝑎𝑖𝑢𝑥0+𝑖,(4.1) where is the spacing of a uniform mesh and the coefficients 𝑎𝑖 are such that 𝑎𝑖=𝑎𝑖, providing a scheme without dissipation. On applying spatial Fourier Transform to (4.1), the effective wavenumber 𝜃 of the scheme is obtained and it is given by𝜃=23𝑖=1𝑎𝑖sin(𝑖𝜃).(4.2)

Taylor expansion of 𝜃 about 𝜃 from (4.2) gives the following:2𝑎11𝜃6(𝜃)3+1120(𝜃)5+2𝑎212𝜃6(2𝜃)3+1120(2𝜃)5+2𝑎313𝜃6(3𝜃)3+1120(3𝜃)5+.(4.3)

To obtain a 4th-order accurate method, we must have2𝑎1+4𝑎2+6𝑎3𝑎=1,1+8𝑎2+27𝑎3=0.(4.4)

Since, we have 2 equations and 3 unknowns, we can choose, for instance, say 𝑎1 as a free parameter. Thus,𝑎2=94205𝑎1,𝑎3=15𝑎123.(4.5)

Hence, the numerical wavenumber can be expressed as𝜃2𝑎19sin(𝜃)+24205𝑎11sin(2𝜃)+25𝑎1215sin(3𝜃).(4.6)

The optimisation procedure used by Tam and Webb [3] is to find 𝑎1 which minimizes the integrated error, 𝐸 defined as𝐸=01.1||𝜃||𝜃2𝑑(𝜃).(4.7) The value obtained for 𝑎1 is 0.7708823806. The corresponding values for 𝑎2 and 𝑎3 are −0.1667059045 and 0.0208431428.

Hence, the scheme developed by Tam and Webb [3] has numerical wavenumber, 𝜃, and group velocity given by𝜃=1.5417647612sin(𝜃)0.3334118090sin(2𝜃)+0.0416862856sin(3𝜃),(4.8)groupvelocity=1.5417647612cos(𝜃)0.6668236180cos(2𝜃)+0.1250588568cos(3𝜃)(4.9) and is termed as “TAM” method.

We next consider the numerical wavenumber in (4.2) and use the technique of MIEELDLD to find optimal values of 𝑎1, 𝑎2, and 𝑎3. The integrated error using MIEELDLD is given by𝐸=01.1||𝜃exp||+||𝜃𝜃||||||𝜃+exp||||𝜃𝜃||||2𝑑(𝜃).(4.10) Since we are considering a 7-point and 4th-order central difference method, the numerical wavenumber, 𝜃, does not have an imaginary part, that is, (𝜃)=0. Hence, (4.10) simplifies to𝐸=01.1||𝜃2exp||𝜃2𝑑(𝜃),(4.11) and on minimising this integral using the function NLPSolve in maple, we obtain 𝑎1 as 0.7677206709. Corresponding values for 𝑎2 and 𝑎3 are 0.1641765367 and 0.0202108009, respectively.

Hence we have obtained a modified method which is 7-point and of 4th-order which we term as “TAM-NEW” method. Expressions for the numerical wavenumber and the group velocity of the “TAM-NEW” method are given by𝜃=1.5354413418sin(𝜃)0.3283530734sin(2𝜃)+0.0404216018sin(3𝜃),(4.12)groupvelocity=1.5354413418cos(𝜃)0.6567061468cos(2𝜃)+0.1212648054cos(3𝜃).(4.13) We next perform a spectral analysis of the two methods. We compare the variation of numerical wavenumber versus the exact wavenumber in Figure 1. A plot of the dispersion error versus the exact wavenumber is depicted in Figure 2. The dispersion error for TAM-NEW is slightly less than that for TAM for 0<𝜃1, but for 1𝜃𝜋/2, the dispersion error from TAM is slightly less than that for TAM-NEW.

783101.fig.001
Figure 1: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods: TAM, TAM-NEW, ZINGG, and ZINGG-NEW.
783101.fig.002
Figure 2: Plot of the dispersion error on a logarithmic scale versus exact wavenumber for the methods: TAM, TAM-NEW, ZINGG, and ZINGG-NEW.

We now compare quantitatively these two methods: TAM and TAM-NEW. We use four accuracy limits [5, 16] defined as follows:||𝜃||𝜃𝜋5×103,||𝜃||𝜃𝜋5×104,||||𝑑𝜃𝑑(𝜃)||||1.05×103,||||𝑑𝜃𝑑(𝜃)||||1.05×104(4.14) and compute the minimum number of points per wavelength needed to resolve a wave for each of the four accuracy limits. The results are summarised in Table 1.

tab1
Table 1: Comparing the dispersive and group velocity properties for two spatial discretisation methods “TAM” and “TAM-NEW” in terms of number of points per wavelength (to 4 d.p).

It is seen that the scheme “TAM-NEW” is not superior to the TAM method as for a given accuracy it requires more points per wavelength in regard to the dispersive and group velocity properties. This is because the technique of MIEELDLD aims to minimize both dispersion and dissipation in numerical methods but here our aim is to construct a 7-point and 4th-order central difference method with no dissipation.

5. Modification to Space Discretisation Scheme Developed by Lockard et al. [13]

Lockard et al. [13] constructed a 7-point and 4th-order difference method by approximating 𝜕𝑢/𝜕𝑥 at 𝑥0 as𝜕𝑢=1𝜕𝑥3𝑖=4𝑎𝑖𝑢𝑥0+𝑖,(5.1) and therefore the real and imaginary parts of the numerical wavenumber are obtained as follows:𝜃=𝑎4sin(4𝜃)𝑎3sin(3𝜃)𝑎2sin(2𝜃)𝑎1sin(𝜃)+𝑎1sin(𝜃)+𝑎2sin(2𝜃)+𝑎3𝜃sin(3𝜃),(5.2)𝑎=4cos(4𝜃)+𝑎3cos(3𝜃)+𝑎2cos(2𝜃)+𝑎1cos(𝜃)+𝑎0+𝑎1cos(𝜃)+𝑎2cos(2𝜃)+𝑎3.cos(3𝜃)(5.3)

To obtain a 4th-order method, we require 4 conditions based on the real and imaginary parts of 𝜃, namely, 𝑎1+2𝑎2+3𝑎34𝑎43𝑎32𝑎2𝑎1=1,𝑎18𝑎227𝑎3+64𝑎4+27𝑎3+8𝑎2+𝑎1𝑎=0,0+𝑎1+𝑎2+𝑎3+𝑎4+𝑎3+𝑎2+𝑎1=0,𝑎14𝑎29𝑎316𝑎49𝑎34𝑎2𝑎1=0.(5.4)

The coefficients obtained by Lockard et al. [13] are𝑎4=0.0103930209,𝑎3=0.0846974943,𝑎2𝑎=0.3420311831,1=1.0526812838,𝑎0=0.2872741244,𝑎1𝑎=0.5861624738,2=0.0981442817,𝑎3=0.0096622576.(5.5)

Hence, the real and imaginary parts of 𝜃 for LOCKARD scheme are given as follows:𝜃=1.63884375sin(𝜃)0.44017538sin(2𝜃)+0.09433201sin(3𝜃)0.01039020sin(4𝜃),(5.6)(𝜃)=0.28727412+0.46651881cos(𝜃)0.24388682cos(2𝜃)+0.07500749cos(3𝜃)0.01039020cos(4𝜃), (23)respectively.

We now obtain a modification to the scheme developed by Lockard et al. [13]. We consider the numerical wavenumber in (5.2) and (5.3) and replace 𝑎1, 𝑎0, 𝑎1, 𝑎2, and 𝑎3 in terms of 𝑎2, 𝑎3, 𝑎4, and 𝜃. Our aim is to minimise the following integral:𝐸=01.1||𝜃exp||+||𝜃𝜃||||||𝜃+exp||||𝜃𝜃||||2𝑑(𝜃).(5.8)

The integral is a function of 𝑎2, 𝑎3, and 𝑎4. We use the function NLPSolve and obtain optimal values for 𝑎4, 𝑎3, and 𝑎2 as 0.0113460667, −0.0891980000, and 0.3499980000. Then the values of the other unknowns can be obtained and we are out with𝑎1=1.0582666667,𝑎0=0.2866010000,𝑎1𝑎=0.5895196001,2=0.1,𝑎3=0.01.(5.9)

The modified method is termed as “LOCKARD-NEW” and has real and imaginary parts of its numerical wavenumber described by𝜃𝜃=1.6477862670sin(𝜃)0.4499980000sin(2𝜃)+0.0991980000sin(3𝜃)0.0113460667sin(4𝜃),(5.10)=0.2866010000+0.4687470669cos(𝜃)0.2499980000cos(2𝜃)+0.0791980000cos(3𝜃)0.0113460667cos(4𝜃),(5.11) respectively.

We next perform a spectral analysis of the two methods: LOCKARD and LOCKARD-NEW. We compare the variation of numerical wavenumber versus the exact wavenumber in Figure 3 and in Figure 4, we have the plot of the dispersion error versus the exact wavenumber.

783101.fig.003
Figure 3: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods LOCKARD and LOCKARD-NEW.
783101.fig.004
Figure 4: Plot of the variation of dispersion error in logarithmic scale versus exact wavenumber for LOCKARD and LOCKARD-NEW schemes.

We now compare quantitatively the two methods by computing the minimum number of points per wavelength needed to resolve a wave for each of the four accuracy limits and the results are summarized in Table 2.

tab2
Table 2: Comparing the dispersive and group velocity properties for two spatial methods LOCKARD and LOCKARD-NEW in terms of number of points per wavelength (to 4 d.p).

Clearly, LOCKARD-NEW has appreciably better phase and group velocity properties as compared to LOCKARD scheme.

6. Modification to Spatial Discretisation Scheme Developed by Zingg et al. [14]

Zingg et al. [14] constructed a 4-point and 4th-order difference method. They approximated 𝜕𝑢/𝜕𝑥 at 𝑥0 by𝜕𝑢=1𝜕𝑥3𝑖=1𝑎𝑖𝑢𝑥0𝑥+𝑖𝑢0+1𝑖𝑑0𝑢𝑥0+3𝑖=1𝑑𝑖𝑢𝑥0𝑥+𝑖+𝑢0.𝑖(6.1)

The real and imaginary parts of the numerical wavenumber are obtained as𝜃𝑎=21sin(𝜃)+𝑎2sin(2𝜃)+𝑎3𝜃sin(3𝜃),(6.2)𝑑=0+2𝑑1cos(𝜃)+2𝑑2cos(2𝜃)+2𝑑3cos(3𝜃).(6.3)

The conditions to have a 4th-order difference method are as follows.

(i)If we consider (𝜃), then 2𝑎1+4𝑎2+6𝑎31=1,6𝑎186𝑎2276𝑎3=0,(6.4) and these two conditions give 𝑎2=94205𝑎1,𝑎(6.5)3=15𝑎123.(6.6)(ii)If we consider (𝜃), then 𝑑0+2𝑑1+2𝑑2+2𝑑3=0,𝑑14𝑑29𝑑3=0,(6.7) and this gives 𝑑0=6𝑑2+16𝑑3,𝑑(6.8)1=4𝑑29𝑑3.(6.9)

Based on the optimisation performed by Zingg et al. [14], the following values are obtained:𝑎1=0.75996126,𝑎2=0.15812197,𝑎3=0.01876090,𝑑0𝑑=0.1,1=0.07638462,𝑑2=0.03228962,𝑑3=0.00590500.(6.10)

We now obtain a modification to the scheme proposed by Zingg et al. [14] using MIEELDLD. We consider𝜃𝑑=0+2𝑑1cos(𝜃)+2𝑑2cos(2𝜃)+2𝑑3cos(3𝜃).(6.11)

Since Im(𝜃) must be negative and the method must have sufficient dissipation, we can choose 𝑑0=0.1 and hence obtain𝑑15=8𝑑2𝑑0.05625,33=0.006258𝑑2.(6.12)

We next plot Im(𝜃) versus 𝑑2 versus 𝜃[0,2𝜋] and obtain the range of 𝑑2 such that Im(𝜃)<0. The maximum value of 𝑑2 is 0.0323. Having fixed the values of 𝑑0 as 0.1 and 𝑑2 as 0.0323, now we can compute the values of 𝑑1 and 𝑑3. We are out with 𝑑1=0.0764375 and 𝑑3=5.8625×103. Hence, we minimize the following integral:01.1||𝜃exp||+||𝜃𝜃||||||𝜃+exp||||𝜃𝜃||||2.0𝑑(𝜃)(6.13) which is a function of 𝑎1, using NLPSolve.

We obtain 𝑎1=0.7643155206, and therefore, using (6.5) and (6.6), we obtain 𝑎2=0.1614524165 and 𝑎3=0.0195297708.

Hence, the real and imaginary parts of the real and imaginary parts of the numerical wavenumber of the scheme ZINGG-NEW are as follows:𝜃𝜃=1.5286310410sin(𝜃)0.3229048330sin(2𝜃)+0.0390595416sin(3𝜃),(6.14)=0.1+0.1528750000cos(𝜃)0.0646000000cos(2𝜃)+0.0117250000cos(3𝜃).(6.15)

Plots of (𝜃) versus 𝜃 and also for (𝜃) versus 𝜃 for ZINGG and ZINGG-NEW schemes are depicted in Figures 1 and 6, respectively. It is observed based on Figure 6 that the two methods have almost the same dissipation error for 𝜃[0,𝜋]. Based on (Figure 1), we observe that for 𝜃<0.2 and 0.8<𝜃<𝜋/2, the dispersion error from ZINGG-NEW is less than that for ZINGG. For 0.2<𝜃<0.8, the dispersion error from ZINGG is less than ZINGG-NEW.

Based on Table 3, for the four accuracy limits tested, we can conclude that the new scheme developed is superior to the ZINGG method in terms of both dispersive and group velocity properties as it requires less points per wavelength in all the four cases.

tab3
Table 3: Comparing the dispersive and group velocity properties for two spatial methods ZINGG and ZINGG-NEW in terms of number of points per wavelength (to 4 d.p).

7. Modification to Spatial Scheme Developed by Zhuang and Chen [15]

Zhuang and Chen [15] constructed a 7-point and 4th-order difference method by approximating 𝜕𝑢/𝜕𝑥 at 𝑥0 as𝜕𝑢=1𝜕𝑥2𝑖=4𝑎𝑖𝑢𝑥0+𝑖,(7.1) and therefore the real and imaginary parts of the numerical wavenumber are obtained as𝜃=𝑎1sin(𝜃)+𝑎2sin(2𝜃)𝑎4sin(4𝜃)𝑎3sin(3𝜃)𝑎2sin(2𝜃)𝑎1𝜃sin(𝜃),(7.2)𝑎=4cos(4𝜃)+𝑎3cos(3𝜃)+𝑎2cos(2𝜃)+𝑎1cos(𝜃)+𝑎0+𝑎1cos(𝜃)+𝑎2.cos(2𝜃)(7.3)

To obtain a 4th-order method, we require 4 conditions based on the real and imaginary parts of 𝜃:𝑎1+2𝑎24𝑎43𝑎32𝑎2𝑎1=1,𝑎18𝑎2+64𝑎4+27𝑎3+8𝑎2+𝑎1𝑎=0,0+𝑎1+𝑎2+𝑎4+𝑎3+𝑎2+𝑎1=0,𝑎116𝑎44𝑎2𝑎14𝑎29𝑎3=0.(7.4)

These simplify to the following if we let 𝑎4,𝑎3,𝑎2 as free parameters:𝑎2=10𝑎4+4𝑎3+𝑎216,𝑎1=20𝑎410𝑎34𝑎213,𝑎0=45𝑎4+20𝑎3+6𝑎212,𝑎1=36𝑎415𝑎34𝑎2+1.(7.5)

On plugging 𝑎2, 𝑎1, 𝑎0, and 𝑎1 in terms of functions of 𝑎4, 𝑎3, 𝑎2 in (7.2) and (7.3), we get𝜃=5165𝑎4+𝑎34115sin(𝜃)+660𝑎4+24𝑎31sin(2𝜃)𝑎3sin(3𝜃)𝑎4𝜃sin(4𝜃),(7.6)=1245𝑎420𝑎36𝑎2+16cos(𝜃)336𝑎4+150𝑎3+48𝑎24𝑎3cos(3𝜃)𝑎41cos(4𝜃)+660𝑎424𝑎312𝑎2+1cos(2𝜃).(7.7)

The coefficients obtained by Zhuang and Chen [15] are𝑎4=0.0161404967,𝑎3=0.1228212790,𝑎2𝑎=0.4553322778,1=1.2492595883,𝑎0=0.5018904380,𝑎1𝑎=0.4399321927,2=0.0412145379,(7.8) and, therefore, the real and imaginary parts of 𝜃 are given as follows:𝜃𝜃=1.689191781sin(𝜃)0.4965468157sin(2𝜃)+0.1228212790sin(3𝜃)0.0161404967sin(4𝜃),(7.9)=0.5018904390+0.8093273950cos(𝜃)0.4141177399cos(2𝜃)+0.1228212790cos(3𝜃)0.0161404967cos(4𝜃),(7.10) respectively.

We now obtain a modification to the scheme developed by Zhuang and Chen [15]. We consider the numerical wavenumber in (7.6) and (7.7) and minimise the following integral𝐸=01.1||𝜃exp||+||𝜃𝜃||||||𝜃+exp||||𝜃𝜃||||2𝑑(𝜃).(7.11)

The integral is a function of 𝑎4, 𝑎3, and 𝑎2. We use the function NLPSolve and obtain optimal values for 𝑎4, 𝑎3, and 𝑎2 as 0.01575, −0.122, and 0.4553 respectively. Corresponding values for 𝑎2, 𝑎1, 𝑎0, and 𝑎1 are then obtained as −0.0418666600, −1.2495333300, 0.5005500000, and 0.4418000000, respectively.

The modified method is termed as ZHUANG-NEW and has real and imaginary parts of its numerical wavenumber described by𝜃𝜃=1.6913333333sin(𝜃)0.4971666667sin(2𝜃)+0.1220000000sin(3𝜃)0.0157500000sin(4𝜃),(7.12)=0.5005500000+0.8077333330cos(𝜃)0.4134333333cos(2𝜃)+0.1220000000cos(3𝜃)0.0157500000cos(4𝜃),(7.13) respectively.

We next perform a spectral analysis of the two methods: ZHUANG and ZHUANG-NEW. We compare the variation of real part and imaginary parts of the numerical wavenumber versus the exact wavenumber in Figures 7 and 5, respectively. We have the plot of the dispersion error versus the exact wavenumber in Figure 8 and we observe that, for 0<𝜃<1, ZHUANG-NEW is slightly better than ZHUANG in terms of dispersive properties.

783101.fig.005
Figure 5: Plot of the imaginary part of numerical wavenumber versus exact wavenumber for the methods: LOCKARD, LOCKARD-NEW, ZHUANG, and ZHUANG-NEW.
783101.fig.006
Figure 6: Plot of the imaginary part of numerical wavenumber versus exact wavenumber for ZINGG and ZINGG-NEW.
783101.fig.007
Figure 7: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods: ZHUANG, ZHUANG-NEW, BOGEY, and BOGEY-NEW.
783101.fig.008
Figure 8: Plot of the variation of dispersion error in logarithmic scale versus exact wavenumber for the methods ZHUANG and ZHUANG-NEW.

We now compare quantitatively these two methods. We compute the minimum number of points per wavelength needed to resolve a wave for each of the four accuracy limits. The results are summarized in Table 4.

tab4
Table 4: Comparing the dispersive and group velocity properties for two spatial methods ZHUANG and ZHUANG-NEW in terms of number of points per wavelength (to 4 d.p).

ZHUANG-NEW requires fewer points per wavelength than ZHUANG scheme for |(𝜃𝜃)/𝜋|0.005.

8. Modification to Spatial Discretisation Scheme Developed by Bogey and Bailly [16]

Bogey and Bailly [16] modified the measure used by Tam and Webb [3] by minimizing the relative difference between 𝜃 and 𝜃. They define the integrated error, 𝐸, as𝐸=𝜋/2𝜋/16||𝜃||𝜃𝜃𝑑(𝜃).(8.1)

Bogey and Bailly [3] use a 9-point stencil with coefficients 𝑎4, 𝑎3, 𝑎2, 𝑎1, 𝑎0, 𝑎1, 𝑎2, 𝑎3, 𝑎4 and choose 𝑎0=0, 𝑎1=𝑎1, 𝑎2=𝑎2, 𝑎3=𝑎3, and 𝑎4=𝑎4 and therefore the numerical wavenumber can be written as𝜃𝑎=21sin(𝜃)+𝑎2sin(2𝜃)+𝑎3sin(3𝜃)+𝑎4sin(4𝜃).(8.2)

To obtain a 4th-order method, 𝑎1 and 𝑎2 are chosen such as𝑎1=23+5𝑎3+16𝑎4,𝑎21=612+24𝑎3+60𝑎4,(8.3) respectively.

The coefficients 𝑎3 and 𝑎4 are chosen to minimize the integrated error defined in (8.1), and the values which Bogey and Bailly [16] have obtained are as follows:𝑎1=0.841570125,𝑎2=0.2446786318,𝑎3=0.0594635848,𝑎4=0.0076509040.(8.4)

We now construct a method based on a 9-point stencil using MIEELDLD. The wavenumber is set as follows:𝜃2=23+5𝑎3+16𝑎41sin(𝜃)+2124𝑎310𝑎4sin(2𝜃)+2𝑎3sin(3𝜃).(8.5)

The integrated error using MIEELDLD is defined as𝜋/2𝜋/16||𝜃2exp||𝜃2𝑑(𝜃),(8.6) which is a function of 𝑎3 and 𝑎4. Using NLPSolve, we obtain the optimal values of 𝑎3 and 𝑎4 as 0.0613000000 and −0.0080500000, respectively. Hence, we obtain 𝑎1 and 𝑎2 as 0.8443666667 and −0.2480333333, respectively.

Using MIEELDLD, a new scheme is obtained and is termed as BOGEY-NEW with its numerical wavenumber given by𝜃=1.6887333332sin(𝜃)0.4960666667sin(2𝜃)+0.1226000000sin(3𝜃)0.0161000000sin(4𝜃).(8.7)

We next perform a spectral analysis of the two methods: BOGEY and BOGEY-NEW. We compare the variation of numerical wavenumber versus the exact wavenumber in Figures 7 and 9; we have the plot of the dispersion error versus the exact wavenumber.

783101.fig.009
Figure 9: Plot of the variation of dispersion error in logarithmic scale versus exact wavenumber.

We now compare quantitatively these two methods. We compute the minimum number of points per wavelength needed to resolve a wave for each of the four accuracy limits.

Table 5 indicates that BOGEY-NEW has appreciably better phase and group velocity properties as compared to BOGEY scheme.

tab5
Table 5: Comparing the dispersive and group velocity properties for two spatial methods BOGEY and BOGEY-NEW in terms of number of points per wavelength (to 4 d.p).

9. Optimized Time Discretisation Schemes

9.1. Time Discretisation Scheme by Tam et al. [17]

Tam et al. [17] have developed a time-marching scheme which is four-level and accurate up to 𝑘3. They expressed𝑈(𝑛+1)𝑈(𝑛)𝑘3𝑗=0𝑏𝑗𝑑𝑈𝑑𝑡(𝑛𝑗).(9.1) We next summarize how the coefficients have been obtained.

The effective angular frequency of the time discretisation method is obtained as𝜛=𝐼(exp(𝐼𝜔𝑘)1)𝑘3𝑗=0𝑏𝑗exp(𝐼𝑗𝜔𝑘).(9.2)

For 𝜛𝑘 to approximate 𝜔𝑘 to order (𝜔𝑘)4, we must have𝑏0+𝑏1+𝑏2+𝑏3𝑏=1,1+2𝑏2+3𝑏31=2,𝑏1+4𝑏2+9𝑏3=13.(9.3)

Since we have 4 equations and 3 unknowns, we can choose 𝑏0 as a free parameter, and hence we have𝑏1=53123𝑏0,𝑏2=3𝑏0163,𝑏3=2312𝑏0.(9.4)

Hence, we can express 𝜛 as follows:𝜛𝑘=𝐴𝐶+𝐵𝐷1+𝐼𝐵𝐶𝐴𝐷1𝐶2+(𝐷1)2,(9.5) where𝐴=sin(𝜔𝑘),𝐵=cos(𝜔𝑘)1,𝐶=𝑏0+53123𝑏0cos(𝜔𝑘)+3𝑏0163cos(2𝜔𝑘)+2312𝑏0𝐷cos(3𝜔𝑘),1=53123𝑏0sin(𝜔𝑘)+3𝑏0163sin(2𝜔𝑘)+2312𝑏0sin(3𝜔𝑘).(9.6)

The weighted integral error incurred by using 𝜛 to approximate 𝜔, used by Tam et al. [17], is computed as𝐸𝑇=0.50.5𝜎((𝜛𝑘)𝜔𝑘)2+(1𝜎)((𝜛𝑘))2𝑑(𝜔𝑘),(9.7) and 𝜎 is chosen as 0.36.

On minimizing 𝐸𝑇, the value of 𝑏0 is obtained as 2.30255809 and therefore the corresponding values for 𝑏1, 𝑏2, and 𝑏3 are −2.49100760, 1.57434094, and −0.38589142, respectively.

9.2. Modified Temporal Discretisation Scheme Using MIEELDLD

We consider the equation in (9.5) which expresses 𝜛𝑘 in terms of 𝜔𝑘 and define the quantity, eeldld as||||+||||||||||||exp(𝜛𝑘)𝜔𝑘(𝜛𝑘)+exp(𝜛𝑘)𝜔𝑘(𝜛𝑘)2.(9.8)

We minimize0.50.5eeldld𝑑(𝜔𝑘)(9.9) and this integral is a function of 𝑏0. Using NLPSolve, we obtain the value of 𝑏0 as 2.2796378228. A plot of 𝐸𝑇 versus 𝑏0 is shown in Figure 10.

783101.fig.0010
Figure 10: Plot of IEELDLD versus 𝑏0.

Corresponding values of 𝑏1,𝑏2,𝑏3 are obtained as −2.4222468020, 1.5055801360, and −0.3629711560. This modified temporal discretisation scheme obtained by modifying the temporal scheme of Tam et al. [17] is termed as “TAM-MODIFIED” scheme. Plots of (𝜛𝑘) versus 𝜔𝑘 and (𝜛𝑘) versus 𝜔𝑘 for the TAM-MODIFIED scheme are shown in Figures 11 and 12, respectively.

783101.fig.0011
Figure 11: Plot of (𝜛𝑘) versus 𝜔𝑘.
783101.fig.0012
Figure 12: Plot of (𝜛𝑘) versus 𝜔𝑘.

For |(𝜛𝑘)|3×103, we require 𝜔𝑘0.42.

9.3. Comparison between Temporal Discretisation Schemes: TAM and TAM-MODIFIED

Plots of (𝜛𝑘) versus 𝜔𝑘 for the two methods are shown in Figure 13. We also compare their dispersive properties at two different levels of accuracy in terms of number of points per wavelength and the results are tabulated in Table 6. Clearly, TAM-MODIFIED is more superior as it requires less points per wavelength for the same accuracy.

tab6
Table 6: Comparing the dispersive properties for two temporal discretisation methods TAM and TAM-MODIFIED in terms of number of points per wavelength (to 4 d.p).
783101.fig.0013
Figure 13: Plot of (𝜛𝑘) versus 𝜔𝑘 for TAM and TAM-MODIFIED schemes.

10. Stability of Some Multilevel Optimized Combined Spatial-Temporal Finite Difference Schemes

The stability of the combined spatial and temporal finite difference scheme developed by Tam and Webb [3] and Tam et al. [17], which is 7-point in space and 4-point in time and which is referred to as the Dispersion-Relation-Preserving (DRP) scheme, satisfies the stability condition, 𝑟0.229 [3]. The condition on the spatial discretisation is that |(𝜃𝜃)/𝜋|0.05 and this gives 𝜃1.76. The interval 0<𝜛𝑘0.4 has been chosen in order to maintain satisfactory temporal resolution and this interval is obtained by requiring the condition: (𝜛𝑘)0.003.

Since𝜛𝑘=(𝛽𝜃)𝑘,(10.1) we also have𝜛𝑘=𝑟𝜃.(10.2)

Since, we require 𝜛𝑘0.4, this implies that 𝑟(𝜃)0.4. Also, we have 𝜃1.76 and thus 𝑟0.4/1.76.

The stability of the DRP scheme therefore satisfies the condition: 𝑟0.23.

Using the approach just described in the preceding paragraph, the ranges of stability of some methods are obtained, namely, TAM-NEW, ZINGG-NEW, ZHUANG-NEW, LOCKARD-NEW, and BOGEY-NEW when combined with TAMMODIFIED. We also obtain the range of stability for the methods: ZINGG, ZINGG, ZHUANG, LOCKARD, and BOGEY when they are combined with the temporal discretisation scheme of Tam et al. [17]. The results are tabulated in Table 7. It is seen that the new combined spatial-temporal methods constructed using MIEELDLD have a slightly greater region of stability than the existing combined spatial-temporal methods.

tab7
Table 7: Region of stability for some combined spatial-temporal discretisation schemes.

11. Comparison of Some Metric Measures

Spatial Scheme of Tam and Webb [3]
The integrated error is defined as 01.1||𝜃||𝜃2𝑑(𝜃).(11.1)
The quantity, |𝜃𝜃|2 is equivalent to |1RPE|2 in a computational fluid dynamics framework. A plot of |1RPE|2 versus RPE[0,2] is shown in Figure 14(a).

fig14
Figure 14: Plot of different metrics from Tam and Webb [3], Bogey and Bailly [16] and Appadu and Dauhoo [18].

Spatial Scheme of Bogey and Bailly [16]
In this case, the integrated error is defined as 𝜋/2𝜋/16||𝜃||𝜃𝜃𝑑(𝜃),(11.2) or ln𝜋/2ln𝜋/16||𝜃||𝑑𝜃(ln(𝜃)).(11.3) The quantity (|𝜃𝜃|)/𝜃 is equivalent to (|1RPE|)/RPE while |𝜃𝜃| is equivalent to |1RPE|. Plots of (|1RPE|)/RPE and |1RPE|, both versus RPE[0,2], are shown in Figures 14(b) and 14(c).

Spatial Scheme Using MIEELDLD
A plot of eeldld=exp((||1RPE|(1AFM)|)+exp(|1RPE|+(1AFM))2) versus RPE[0,2] versus AFM[0,1] is shown in Figure 14(d).
We observe from Figures 14(a), 14(b), and 14(c) that the measure is zero when RPE=1 whereas, in Figure 14(d), the measure is zero provided RPE=1 and AFM=1.

12. Conclusions

In this work, we have used the technique of Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) in a computational aeroacoustics framework to obtain modifications to optimized spatial schemes constructed by Tam and Webb [3], Zingg et al. [14], Lockard et al. [13], Zhuang and Chen [15], and Bogey and Bailly [16], and also a modification to the optimized temporal scheme devised by Tam et al. [17] is obtained. It is seen that, in general, improvements can be made to the existing spatial discretisation schemes, using MIEELDLD. The new temporal scheme obtained using MIEELDLD is superior in terms of dispersive properties as compared to the one constructed by Tam et al. [17]. The region of stability has also been obtained. In a nutshell, we conclude that MIEELDLD is an efficient technique to construct high order methods with low dispersion and dissipative properties. An extension of this work will be to use the new spatial discretisation schemes and the novel temporal discretisation method constructed to solve 1-D wave propagation experiments and quantify the errors into dispersion and dissipation. Moreover, MIEELDLD can be used to construct low dispersive and low dissipative methods which approximate 2-D and 3-D scalar advection equation suited for computational aeroacoustics applications.

Nomenclature

𝐼=(1)
𝑘:Time step
:Spatial step
𝑛:Time level
𝛽:Advection velocity
𝜃:Numerical wavenumber
𝜃:Exact wavenumber
𝑟:c/courant number
𝑟=𝛽𝑘/
𝑤:Phase angle in 1-D
𝑤=𝜃
𝜔:Exact angular frequency
𝜛:Effective angular frequency of time discretization scheme
RPE:Relative phase error per unit time step
AF:Amplification factor
AFM=|𝐴𝐹|
LDLD:Low Dispersion and Low Dissipation
IEELDLD:Integrated Exponential Error for Low Dispersion and Low Dissipation
MIEELDLD:Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation.

Acknowledgment

This work was funded through the Research Development Programme of the University of Pretoria.

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