The Technique of MIEELDLD in Computational Aeroacoustics

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 (cid:2) Appadu and Dauhoo, 2011; Appadu, 2012a; Appadu, 2012b; Appadu, 2012c (cid:3) . This technique has been baptised as the Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (cid:2) MIEELDLD (cid:3) 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 (cid:2) 1993 (cid:3) , Lockard et al. (cid:2) 1995 (cid:3) , Zingg et al. (cid:2) 1996 (cid:3) , Zhuang and Chen (cid:2) 2002 (cid:3) , and Bogey and Bailly (cid:2) 2004 (cid:3) have been obtained, and also a modiﬁcation to the temporal scheme developed by Tam et al. (cid:2) 1993 (cid:3) has been obtained. These novel methods obtained using MIEELDLD have in general better dispersive properties as compared to the existing optimised methods. and the two-step Lax-Friedrichs developed by Liska and Wendro ﬀ (cid:4) 25 (cid:5) has been computed and some numerical experiments have been performed such as 2-D solid body rotation test (cid:4) 26 (cid:5) , 2-D acoustics (cid:4) 27 (cid:5) , and 2-D circular Riemann problem (cid:4) 26 (cid:5) . We have shown that better results are obtained when the optimal parameters obtained using MIEELDLD are used. and Minimised Integrated Error from Berland et al. (cid:4) 28 (cid:5) (cid:2) MIEBERLAND (cid:3) respectively. It is shown that MIEELDLD has an upper hand over the other techniques of optimisation: MIETAM, MIEBOGEY, and MIEBERLAND. composite methods using and Beam-Warming as a and Beam-Warming schemes or MacCormack and two-step Lax-Friedrichs and the composite MacCormack/Lax-Friedrichs


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 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 θh θ * h − 1.0 which must be almost one in order to reproduce exact result 12 .Otherwise, dispersive patterns appear.When Re d/ d θh θ * h 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.

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 4-8, 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.

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: Then the modulus of the Amplification Factor AFM and the relative phase error RPE are calculated as where r and w are the cfl number and phase angle, respectively.For a scheme to have Low Dispersion and Low Dissipation, we require The quantity, |1−RPE| measures dispersion error while 1−AFM measures dissipation error.Also when dissipation neutralises dispersion optimally, we have 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: Similarly, we expect 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 for a range of w ∈ 0, w 1 , 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 λ : 1 − λ.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 eeldld dw dr, 3.9 which will be in terms of λ.
The value of r 1 is chosen to suit the region of stability of the numerical scheme under consideration while w 1 is chosen such that the approximated RPE ≥ 0 for r ∈ 0, r 1 .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 eeldld dw 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, r 1 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 ,  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

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 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 .

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 x 0 as ∂u ∂x where h is the spacing of a uniform mesh and the coefficients a i are such that a i −a −i , providing a scheme without dissipation.On applying spatial Fourier Transform to 4.1 , the effective wavenumber θ * h of the scheme is obtained and it is given by Taylor expansion of θ * h about θh from 4.2 gives the following: To obtain a 4th-order accurate method, we must have 2a 1 4a 2 6a 3 1, a 1 8a 2 27a 3 0.

4.4
Since, we have 2 equations and 3 unknowns, we can choose, for instance, say a 1 as a free parameter.Thus,

4.5
Hence, the numerical wavenumber can be expressed as The optimisation procedure used by Tam and Webb 3 is to find a 1 which minimizes the integrated error, E defined as The value obtained for a 1 is 0.7708823806.The corresponding values for a 2 and a 3 are −0.1667059045 and 0.0208431428.Hence, the scheme developed by Tam and Webb 3 has numerical wavenumber, θ * h, and group velocity given by θ * h 1.5417647612 sin θh − 0.3334118090 sin 2θh 0.0416862856 sin 3θh , 4.8 groupvelocity 1.5417647612 cos θh − 0.6668236180 cos 2θh 0.1250588568 cos 3θh 4.9 and is termed as "TAM" method.

Journal of Applied Mathematics
We next consider the numerical wavenumber in 4.2 and use the technique of MIEELDLD to find optimal values of a 1 , a 2 , and a 3 .The integrated error using MIEELDLD is given by 4.10 Since we are considering a 7-point and 4th-order central difference method, the numerical wavenumber, θ * h, does not have an imaginary part, that is, θ * h 0. Hence, 4.10 simplifies to and on minimising this integral using the function NLPSolve in maple, we obtain a 1 as 0.7677206709.Corresponding values for a 2 and a 3 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 by θ * h 1.5354413418 sin θh − 0.3283530734 sin 2θh 0.0404216018 sin 3θh , 4.12 groupvelocity 1.5354413418 cos θh − 0.6567061468 cos 2θh 0.1212648054 cos 3θh .

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 < θh ≤ 1, but for 1 ≤ θh ≤ π/2, the dispersion error from TAM is slightly less than that for TAM-NEW.We now compare quantitatively these two methods: TAM and TAM-NEW.We use four accuracy limits 5, 16 defined as follows: 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.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.

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 x 0 as ∂u ∂x and therefore the real and imaginary parts of the numerical wavenumber are obtained as follows: To obtain a 4th-order method, we require 4 conditions based on the real and imaginary parts of θ * h, namely,

5.4
The coefficients obtained by Lockard et  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 a −1 , a 0 , a 1 , a 2 , and a 3 in terms of a −2 , a −3 , a −4 , and θh.Our aim is to minimise the following integral: 5.8 The integral is a function of a −2 , a −3 , and a −4 .We use the function NLPSolve and obtain optimal values for a −4 , a −3 , and a −2 as 0.0113460667, −0.0891980000, and 0.3499980000.Then the values of the other unknowns can be obtained and we are out with a −1 −1.0582666667, a 0 0.2866010000, a 1 0.5895196001, a 2 −0.1, a 3 0.01.

5.9
The modified method is termed as "LOCKARD-NEW" and has real and imaginary parts of its numerical wavenumber described by 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.
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.
Clearly, LOCKARD-NEW has appreciably better phase and group velocity properties as compared to LOCKARD scheme.

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 x 0 by ∂u ∂x

6.3
The conditions to have a 4th-order difference method are as follows.Since Im θ * h must be negative and the method must have sufficient dissipation, we can choose d 0 0.1 and hence obtain

6.12
We next plot Im θ * h versus d 2 versus θh ∈ 0, 2π and obtain the range of d 2 such that Im θ * h < 0. The maximum value of d 2 is 0.0323.Having fixed the values of d 0 as 0.1 and d 2 as 0.0323, now we can compute the values of d 1 and d 3 .We are out with d 1 −0.0764375 and d 3 −5.8625× 10 −3 .Hence, we minimize the following integral: which is a function of a 1 , using NLPSolve.We obtain a 1 0.7643155206, and therefore, using 6.5 and 6.6 , we obtain a 2 −0.1614524165 and a 3 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: θ * h 1.5286310410 sin θh − 0.3229048330 sin 2θh 0.0390595416 sin 3θh , 6.14 θ * h −0.1 0.1528750000 cos θh − 0.0646000000 cos 2θh 0.0117250000 cos 3θh .

6.15
Plots of θ * h versus θh and also for θ * h versus θh 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 θh ∈ 0, π .Based on Figure 1 , we observe that for θh < 0.2 and 0.8 < θh < π/2, the dispersion error from ZINGG-NEW is less than that for ZINGG.For 0.2 < θh < 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.

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 x 0 as ∂u ∂x   To obtain a 4th-order method, we require 4 conditions based on the real and imaginary parts of θ * h:

7.4
These simplify to the following if we let a −4 , a −3 , a −2 as free parameters:

7.5
On plugging a 2 , a −1 , a 0 , and a 1 in terms of functions of a −4 , a −3 , a −2 in 7.2 and 7.3 , we get

7.7
The coefficients obtained by 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 integral The integral is a function of a −4 , a −3 , and a −2 .We use the function NLPSolve and obtain optimal values for a −4 , a −3 , and a −2 as 0.01575, −0.122, and 0.4553 respectively.Corresponding values for a 2 , a −1 , a 0 , and a 1 are then obtained as −0.0418666600, −1.2495333300, 0.5005500000, and 0.4418000000, 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 < θh < 1, ZHUANG-NEW is slightly better than ZHUANG in terms of dispersive properties.
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.

8.2
To obtain a 4th-order method, a 1 and a 2 are chosen such as respectively.The coefficients a 3 and a 4 are chosen to minimize the integrated error defined in 8.1 , and the values which Bogey and Bailly 16 have obtained are as follows: We now construct a method based on a 9-point stencil using MIEELDLD.The wavenumber is set as follows: The integrated error using MIEELDLD is defined as which is a function of a 3 and a 4 .Using NLPSolve, we obtain the optimal values of a 3 and a 4 as 0.0613000000 and −0.0080500000, respectively.Hence, we obtain a 1 and a 2 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 by θ * h 1.6887333332 sin θh − 0.4960666667 sin 2θh 0.1226000000 sin 3θh 0.0161000000 sin 4θh .

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.
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.

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 k 3 .They expressed We next summarize how the coefficients have been obtained.The effective angular frequency of the time discretisation method is obtained as For k to approximate ωk to order ωk 4 , we must have

9.6
The weighted integral error incurred by using to approximate ω, used by Tam et al. 17 , is computed as and σ is chosen as 0.36.
On minimizing E T , the value of b 0 is obtained as 2.30255809 and therefore the corresponding values for b 1 , b 2 , and b 3 are −2.49100760,1.57434094, and −0.38589142, respectively.

Modified Temporal Discretisation Scheme Using MIEELDLD
We consider the equation in 9.5 which expresses k in terms of ωk and define the quantity, eeldld as

Comparison between Temporal Discretisation Schemes: TAM and TAM-MODIFIED
Plots of k 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.

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 The stability of the DRP scheme therefore satisfies the condition: r ≤ 0.23.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.

Comparison of Some Metric Measures
Spatial Scheme of Tam and Webb [3] The integrated error is defined as 11.1 The quantity, |θ * h − θh| 2 is equivalent to |1 − RPE| 2 in a computational fluid dynamics framework.A plot of |1 − RPE| 2 versus RPE ∈ 0, 2 is shown in Figure 14 a .

Journal of Applied Mathematics
Spatial Scheme of Bogey and Bailly [16] In this case, the integrated error is defined as
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.

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.

Figure 1 :
Figure 1: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods: TAM, TAM-NEW, ZINGG, and ZINGG-NEW.

Figure 2 :
Figure 2: Plot of the dispersion error on a logarithmic scale versus exact wavenumber for the methods: TAM, TAM-NEW, ZINGG, and ZINGG-NEW.

Figure 3 :Figure 4 :
Figure 3: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods LOCKARD and LOCKARD-NEW.

Figure 5 :
Figure 5: Plot of the imaginary part of numerical wavenumber versus exact wavenumber for the methods: LOCKARD, LOCKARD-NEW, ZHUANG, and ZHUANG-NEW.

Figure 6 :
Figure 6: Plot of the imaginary part of numerical wavenumber versus exact wavenumber for ZINGG and ZINGG-NEW.

Figure 7 :
Figure 7: Plot of the variation of numerical wavenumber versus exact wavenumber for the methods: ZHUANG, ZHUANG-NEW, BOGEY, and BOGEY-NEW.

8 . 16 ]Figure 8 :
Figure 8: Plot of the variation of dispersion error in logarithmic scale versus exact wavenumber for the methods ZHUANG and ZHUANG-NEW.

Figure 9 :
Figure 9: Plot of the variation of dispersion error in logarithmic scale versus exact wavenumber.
is a function of b 0 .Using NLPSolve, we obtain the value of b 0 as 2.2796378228.A plot of E T versus b 0 is shown in Figure 10.Corresponding values of b 1 , b 2 , b 3 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 k versus ωk and k versus ωk for the TAM-MODIFIED scheme are shown in Figures 11 and 12, respectively.For | k | ≤ 3 × 10 −3 , we require ωk ≤ 0.42.

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

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 .
al. 13 are

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 .

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 .

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 .

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 .

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 .
Integrated Exponential Error for Low Dispersion and Low Dissipation MIEELDLD: Minimised Integrated Exponential Error for Low Dispersion and LowDissipation.