An overview of high-order finite difference schemes for computational aeroacoustics W. De Roeck, W. Desmet, M. Baelmans, P. Sas K.U.Leuven, Department of Mechanical Engineering Celestijnenlaan 300 B, B-3001, Leuven, Belgium e-mail:
[email protected]
Abstract One of the problems in computational aeroacoustics (CAA) is the large disparity between the length and time scales of the flow field, which may be the source of aerodynamically generated noise, and the ones of the resulting acoustic field. This is the main reason why numerical schemes, used to calculate the timeand space-derivatives, should exhibit a low dispersion and dissipation error. This paper focuses on the evaluation of a number of numerical schemes. The methods that are included, are a representative selection of the most commonly used numerical schemes in CAA. Four different spatial schemes are analyzed:(1) a standard 7-point central difference scheme,(2) a standard 9-point central difference scheme,(3) the Dispersion-Relation-Preserving scheme and (4) a 9-point optimized central difference scheme. For the time integration, six different Runge-Kutta methods are analyzed:(1) a standard 5-stage Runge-Kutta,(2) a 5-stage optimized Runge-Kutta,(3) the 5-stage low-dispersion low-dissipation Runge-Kutta,(4) a standard 6stage Runge-Kutta,(5) a 6-stage optimized Runge-Kutta and (6) the 6-stage low-dispersion low-dissipation Runge-Kutta. The different methods are tested for a 1D-propagation problem.
1
Introduction
Computational Aeroacoustics (CAA) focuses on the generation and propagation of aerodynamically generated noise. There are a number of differences between the noise generating flow field and the resulting acoustic field, which make the use of accurate numerical schemes, with a minimum of dispersion and dissipation, necessary [1]: • Aeroacoustic noise is broadband, and the spectrum can be fairly wide. The spatial resolution of the numerical scheme is dictated by the sound waves with the shortest wavelengths. Typically a minimum of six to eight grid-points per wavelength is required. The development of finite difference algorithms that give an adequate solution with a minimum number of grid points per wavelength is an important issue. • The amplitude of the acoustic variables is several orders of magnitude smaller than those of the noise generating aerodynamic flow field. This disparity between acoustic and flow variables inquires accurate numerical schemes. The computational error, introduced by the numerical algorithm, must be smaller than the amplitude of the acoustic variables. Otherwise the solution may be prohibitively corrupted by numerical noise. • The length scales of the noise generating mechanisms in a flow are typically smaller than the wavelengths of the generated sound waves. In the flow field there exists a large disparity between the length scales: going from the smallest scale of turbulence, the Kolmogorov scale, to the size of the largest eddy. For the relation between the necessary grid size of the acoustic field (∆xac ) and that of the flowfield (∆xf l ) the following rule of thumb is often used: ∆xac = ∆xf l /M with M = u0 /c0 the mean 353
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Mach number of the flow and c0 the speed of sound. Finite difference methods become unstable when the CFL-number (CFL = c0 ∆t/∆x) is larger than a critical value. It follows that the computational timestep (∆t) is dictated by the smallest grid size. To avoid too excessive calculation times, a stability range as large as possible is required. • Acoustic waves may propagate over long distances and in all directions. The computed solution must be uniformly accurate over the whole propagation distance. For this reason, the numerical scheme must exhibit a minimum of dispersion and dissipation. If a large number of grid points is used, this is not difficult to achieve but since a minimum number of grid points per wavelength is aimed at in order to keep the calculation times within reasonable limits, this is an issue of utmost importance. Several numerical schemes are more or less suitable for CAA [2] . For the calculation of the space derivatives, standard high-order central difference schemes can be used. Tam et al. [3] and Bogey et al. [4] have derived central difference schemes with low dispersive properties. Time integration in CAA is usually done with higher-order Runge-Kutta schemes. In order to improve the dissipation and dispersion characteristics of these schemes, several other schemes were proposed (e.g. Bogey et al. [4], Hu et al. [5]). In this paper following schemes are evaluated: a standard 6th - and 8th -order central difference scheme (ST7,ST9) and the schemes of Tam (DRP) and Bogey (STO9) for the space integration and a classical 5-stage (RK5) and 6-stage (RK6) Runge Kutta-scheme together with the optimized schemes by Bogey et al.(RKO5, RKO6) and Hu et al. (LddRK5, LddRK6). In the first part of this paper, the basic concepts of the numerical schemes for space and time integration are discussed. In the second part the different numerical schemes are used for solving the one-dimensional convective wave equation. For the different numerical schemes the stability range, the minimum number of grid points per wavelength and the dispersion and dissipation errors are evaluated. A comparison between the different methods is made in the last part of this paper.
2 Basic concepts of space- and time-integration schemes 2.1 2.1.1
Space integration Standard central differences
In the finite difference method, a spatial derivative of a function in a grid point is approximated by a weighted summation of the function values in the neighbouring grid points. For a symmetric (2N + 1)-point stencil, with an equidistant grid spacing (∆x), one can write: N ∂f 1 X (x) ≈ aj f (x + j∆x) ∂x ∆x j=−N
(1)
The coefficients aj can be chosen to obtain a scheme with a maximal accuracy, a minimal dispersion and dissipation or a combination of both. The standard way to create a central difference scheme is to use a Taylor Series truncation. For each of the neighbouring grid points, a Taylor Series expansion can be written. The coefficients aj can be chosen in such a way that a number of higher-order terms are eliminated. In this way, a numerical scheme with a maximal accuracy of (2N )th -order can be obtained with a (2N + 1)-point symmetric stencil. For a 7-point (ST7) and 9-point (ST9) stencil, a 6th - resp. 8th -order accurate scheme is obtained with the coefficients of Table 1. The more points used in the stencil, the more accurate becomes the scheme. The main drawback of this central differences based on the Taylor Series truncation method is the fact that they are not constructed from a dispersive perspective. By using a finite difference approximation for the space derivatives, the propagation
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a1 a2 a3 a4
a0 = −a−1 = −a−2 = −a−3 = −a−4
ST7 0 45/60 −9/60 1/60 -
ST9 0 224/280 −56/280 32/840 −1/820
Table 1: aj for the ST7 and ST9 scheme properties of the partial differential equation (PDE) are changed. Applying a spatial Fourier transformation to (1), one obtains the following relation with antisymmetric coefficients aj (a0 = 0 and aj = −a−j ):
ik f˜ (x) ≈
N X
2i aj sin (jk∆x) f˜ (x) ∆x j=1
(2)
with k the physical wavenumber of the PDE. Comparing the left and the right part of equation (2), one obtains an effective wavenumber k ∗ of the finite difference scheme: N X
k∆x ≈ k ∗ ∆x = 2
aj sin (jk∆x)
(3)
j=1
k ∗ ∆x is a periodic function of k∆x with a period of 2π. Due to the difference between the physical and the numerical wavenumber some wave components propagate faster or slower than the wave speed of the original PDE and dispersion errors are induced. It will be shown that it is possible to construct central difference schemes which are of lower order than the schemes based on the Taylor Series truncation method but which exhibit smaller dispersion errors. 2.1.2
Dispersion-relation-preserving scheme
Tam and Webb [3] constructed a 7-point 4th -order central difference scheme based on a minimalisation of the dispersion error. To assure a minimal dispersion, the coefficients aj of equation (1) can be chosen to minimize the integrated error E of the difference between the physical wavenumber k∆x and the effective wavenumber k ∗ ∆x over a certain wavenumber range ([(k∆x)l , (k∆x)h ]). The boundaries (k∆x)l and (k∆x)h are chosen to obtain a minimal dispersion error in this interval: Z
(k∆x)h
E=
|k∆x − k ∗ ∆x|2 d (k∆x)
(4)
(k∆x)l
Tam and Webb combined the Taylor Series truncation method with the Fourier Transform Optimization method and constructed a 7-point central difference scheme which they called the Dispersion-RelationPreserving scheme (DRP). They choose the coefficients a2 and a3 to obtain a 4th -order accurate scheme. The coefficient a1 was chosen to minimize the integrated error E (with (k∆x)l = 0 and (k∆x)h = 1.1): ∂E =0 ∂a1 The different coefficients for the DRP-scheme are given in Table 2. 2.1.3
(5)
Optimized 9-point stencil central difference scheme
Bogey and Bailly [4], using the same theory as Tam and Webb, do not minimize the absolute difference between k∆x and k ∗ ∆x but the relative difference. The integrated error E then becomes: Z
(k∆x)h
E= (k∆x)l
|k∆x − k ∗ ∆x| d (k∆x) = k∆x
Z
ln(k∆x)h
ln(k∆x)l
|k∆x − k ∗ ∆x| d (ln (k∆x))
(6)
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a0 a1 = −a−1 a2 = −a−2 a3 = −a−3
0 0.77088238051822552 −0.166705904414580469 0.02084314277031176
Table 2: aj for the DRP scheme Bogey and Bailly use a 9-point stencil with antisymmetric coefficients. The coefficients a3 and a4 are chosen with the Taylor Series truncation method to obtain a 4th -order accurate scheme. The two remaining coefficients a1 and a2 are chosen to minimize the integrated error of equation (6) (with (k∆x)l = π/16 and (k∆x)h = π/2). The result is a 9-point, 4th -order central difference scheme (STO9) with the coefficients of Table 3. a1 a2 a3 a4
a0 = −a−1 = −a−2 = −a−3 = −a−4
0 0.841570125482 −0.244678931765 0.059463584768 −0.007650904064
Table 3: aj for the STO9 scheme
2.1.4
Short wave components
Figure 1 plots the effective wave number k ∗ ∆x of each spatial scheme against the true physical wavenumber k∆x. For long waves (k ∗ ∆x < ±1.2) k ∗ is a good approximation for k for all schemes. However at higher frequencies, k ∗ begins to differ from k. An important consequence of this discrepancy is numerical dispersion. The short waves are highly dispersive and propagate with a speed quite different from the physical wave speed of the original PDE. These short waves can cause spurious high-frequency oscillations and can lead to instabilities [6]. Short waves can be generated by discontinuous initial conditions, nonhomogeneities as well as by reflections of acoustic waves generated at the boundaries of the computational domain or at grid interfaces.
Figure 1: k∗ ∆x vs. k∆x for the different spatial finite difference schemes(-=ST7, - - -=DRP, ...=ST9, -.-=STO9) Tam et al. [6] use the concept of group velocity to classify the different wavecomponents. They show that the nondimensional group velocity of the finite difference equations is nearly equal to the derivative ∂k ∗ /∂k.
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The nondimensional group velocities are plotted in figure 2. It is clear that the low-frequency waves are nearly non-dispersive, having a group velocity which is almost equal to the propagation speed of the PDE. However, oen must notice that the dimensionless group velocity is not strictly equal to one, this is especially the case for the STO9-scheme. The differences may appear small but over long propagation distances the cumulative effect may become significant.
Figure 2: Nondimensional group velocity of the spatial finite difference schemes (left: whole frequency range, right: zoom view)(-=ST7, - - -=DRP, ...=ST9, -.-=STO9)
Tam et al. classified the waves (for the DRP-scheme) into three categories: • Long Waves (k∆x ≤ 1.2) which are propagated through the medium almost without dispersion. • Dispersive Waves (1.2 ≤ k∆x ≤ 2.0) which have a positive non-dimensional group velocity, smaller than 1, and are quite dispersive. • Parasite Waves (2.0 ≤ k∆x) which have large wavenumbers and hence very short wavelengths. They are responsible for grid-to-grid oscillations which propagate with a velocity of sometimes more than twice that of the long waves. These are the most dispersive waves. The dispersive and parasite waves are spurious. To improve the quality of the solution, they should be removed from the computation as soon as they are generated. Tam et al. propose to implement this strategy by adding artificial damping to the finite difference scheme. This can be done by adding a damping term to the original PDE: ∂U (x) = F (U (x)) + Du (x) (7) ∂t where F (U ) is a function of the unknown (U ) and its spatial the derivatives and Du (x) the artificial damping term. The damping should selectively damp out the short waves but should have a minimal effect on the long waves. The damping terms of each grid point can be assumed to be linear and proportional to the value of the variables in the neighbouring grid points: Du (x) = −µ
N X
cj u(x + j∆x)
(8)
j=−N
with µ a damping constant, cj the damping coefficients, and u the variable which should be damped. The damping function should be symmetric (cj = c−j ). A proper way to obtain the damping coefficients is to apply a Fourier transformation to eq.(8):
˜ u (k∆x) = c0 + 2 ∗ D
N X j=1
cj cos(jk∆x)
(9)
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˜ u (k∆x) should be normalized (D ˜ u (0) = 0 and D ˜ u (π) = 1.0) and the difference with an ideal damping D ˜ function Did (k∆x) can be minimized: ˜ u (k∆x)−D ˜ id |2 R π |D
∂
0
(k∆x)
d(k∆x) =0
∂cj
j = 1...N
(10)
Tam et al. proposed to use a Gaussian function as an ideal damping function: (
˜ id (k∆x) = exp −ln(2) k∆x − π D σ
2 )
(11)
where σ is the halfwidth of the Gaussian function. In this paper three different damping functions are compared (Fig. 3): one with a Gaussian function with σ = 0.3π (D1), one with σ = 0.2π (D2) and one function with a Gaussian-like shape but a much larger half-width (D3). The coefficients cj and the different damping functions are given in table 4 for a 7-point stencil.
c0 c1 = c−1 c2 = c−2 c3 = c−3
D1 0.3276986608 −0.235718815 0.0861506696 −0.0142811847
D2 0.287392842460 −0.226146951809 0.106303578770 −0.023853048191
D3 0.49 −0.285 0.005 0.035
Table 4: cj for the different damping functions
Figure 3: Different Damping curves(...=D1, - - -=D2, -.-=D3)
2.2
Time integration
Runge-Kutta schemes are the most commonly used type of time advancing schemes in CAA. Only these methods are considered in the present paper. The time evolution equation can be written as ∂U = F (U ) (12) ∂t in which U represents the vector containing the solution values at the spatial grid and the operator F contains the discretization of the spatial derivatives. An explicit low storage p-stage Runge-Kutta scheme advances the solution from time level tn to tn + ∆t as u0 = un ul = un + αl ∆tF (ul−1 ) u
n+1
= u
p
for l = 1, . . . , p
(13)
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The standard p-stage Runge-Kutta schemes of pth -order (for a linear operator F ) can be obtained with Taylor Series expansion of u(tn + ∆t). The coefficients αj for a standard 5- and 6- stage Runge-Kutta scheme (RK5 and RK6) are written in Table 5. These coefficients give the maximum order of accuracy which can be
α1 α2 α3 α4 α5 α6
RK5 1/5 1/4 1/3 1/2 1 -
RK6 1/6 1/5 1/4 1/3 1/2 1
Table 5: αj for RK5 and RK6 obtained with a 5- or 6- stage Runge-Kutta scheme. Like with spatial integration, the coefficients αj can also be chosen to minimize the dispersion and dissipation errors. Hu et al. [5] consider the amplification factor r to analyze the numerical errors in the Runge-Kutta schemes. Equation (13) can be written as un+1 = un +
p X
p Y
αl ∆tj
j=1 l=p−j+1
|
{z γj
∂ j un ∂tj
(14)
}
Applying a temporal Fourier transformation to this equation, the amplification factor of the algorithm is given by: p X u ˜n+1 (ω) ∗ r(ω∆t) = n =1+ γj (iω∆t)j = |r∗ (ω∆t)| eiω ∆t (15) u ˜ (ω) j=1 with |r∗ (ω∆t)| the amplification rate and iω ∗ ∆t the effective angular frequency. The exact amplification factor re is found to be re (ω∆t) = eiω∆t
(16)
Comparing the numerical and the exact amplification factors, the amount of dissipation can be written as 1 − |r∗ (ω∆t)| and the dispersion error (or phase difference) as ω∆t − ω ∗ ∆t. Hu et al. [5] minimize the error between the exact and the numerical solution by determining the coefficients γj , in such a way that the following integral is minimal. The boundaries (ω∆t)h and (ω∆t)l are chosen in such a way that there is a minimal dispersion in this interval. Z
(ω∆t)h
(ω∆t)l
2 p X j iω∆t 1 + γj (iω∆t) − e d(ω∆t) j=1
(17)
For a 5-stage Runge-Kutta scheme (LddRK5), Hu et al. combine the Taylor series expansion method to obtain a 2nd -order accurate schemes (α5 , α4 ) with the Fourier transform optimization method to obtain a minimal dispersion and dissipation (α3 , α2 , α1 )(with (ω∆t)h = 0 and (ω∆t)h ≈ π/2). In this way, they obtain a 5-stage Runge-Kutta scheme with low dispersion and low dissipation properties. The same can be done for a 6-stage Runge-Kutta scheme with a 4th order accuracy (LddRK6). The coefficients αj of both schemes are given in table 6. Bogey and Bailly [4] use the same technique as Hu et al., but they calculate the relative dissipation and the dispersion errors separately and obtained the coefficients γj by minimizing following integrals (with
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α1 α2 α3 α4 α5 α6
LddRK5 0.197707993 0.237179241 0.333116 1/2 1 -
LddRK6 0.169193539 0.1874412 1/4 1/3 1/2 1
Table 6: αj for LddRK (ω∆t)l = π/16 and (ω∆t)h = π/2): Z
ln(ω∆t)h
∗
|1 − r (ω∆t)| d ln(ω∆t) +
Z
ln(ω∆t)h
ln(ω∆t)l
ln(ω∆t)l
|ω ∗ ∆t − ω∆t| d ln(ω∆t) π
(18)
Combining the Taylor series expansion method with the Fourier transform optimization method, they obtain the coefficients of table 7 for a 5-stage Runge-Kutta scheme with 2nd -order accuracy (RKO5) and a 6-stage Runge-Kutta scheme with 2nd -order accuracy (RKO6). α1 α2 α3 α4 α5 α6
RKO5 0.181575486 0.238260222 0.330500707 1/2 1 -
RKO6 0.117979902 0.184646967 0.246623604 0.331839543 1/2 1
Table 7: αj for RKO5 Figure 4 gives an overview of the dissipation error (|1 − |r∗ (ω∆t)||) and dispersion error (|ω ∗ ∆t − ω∆t|) of the different Runge-Kutta schemes.
Figure 4: Dissipation (left) and dispersion (right) error of the different Runge-Kutta schemes)(red: -=RK5, - - -=RKO5, -.=LddRK5, blue: -=RK6, - - -=RKO6, -.-=LddRK6)
2.3 Stability considerations As already mentioned before, there is a difference between the effective wave number k ∗ ∆x of each numerical spatial scheme and the true physical wavenumber k∆x. If one imposes a maximum difference
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δ (δ = |k ∗ ∆x − k∆x|) between these two values, one can calculate the minimal resolution (points-perwavelength) of the numerical scheme (resmin = 2π/k ∗ ∆x). The minimal resolution of the different spatial schemes and different values of δ are given in table 8. δ 0.05 0.01 0.005 0.001 0.0005 0.0001 0.00005
ST5 k ∗ ∆x 1.39 1.09 0.98 0.77 0.69 0.55 0.5
resmin 4.5 5.8 6.4 8.2 9.1 11.4 12.6
DRP k ∗ ∆x 1.5 1.25 1.17 1.06 0.64 0.45 0.37
resmin 4.2 5.0 5.4 5.9 9.8 14.0 17.0
ST9 k ∗ ∆x 1.58 1.29 1.19 0.98 0.90 0.75 0.69
resmin 4.0 4.9 5.3 6.4 7.0 8.4 9.1
STO9 k ∗ ∆x 1.8 1.61 1.55 1.23 0.78 0.47 0.4
resmin 3.5 3.9 4.1 5.1 8.1 13.4 15.7
Table 8: Resolution (points-per wavelength) of the different spatial schemes and different values of δ The value of ∆x must be chosen first in order to obtain an accurate result up to the desired frequency or wavelength. The number of points per wavelength must be high enough. Based on this value ∆x a value for the timestep ∆t can be chosen. The choice of the timestep ∆t is an important issue for the Runge-Kutta schemes. Runge-Kutta methods become unstable if ∆t is too large in comparison with the space step ∆x. A first criterium to choose ∆t is that the time advancing scheme is stable. Classical Runge-Kutta schemes are stable if ω∆t ≤ R (with R ≈ 3.54 for the 5-stage and R ≈ 1.75 for the 6-stage Runge Kutta method) [5]. ω is dependent of the effective wavenumber of the spatial difference scheme k ∗ (ω = c0 k ∗ ). In this way, one can write an expression for the CFL-number of the numerical scheme. ∆t R ≤ ∗ (19) ∆x k ∆x with the values for k ∗ ∆x of table 8 the maximal CFL-number of the different numerical schemes for classical 4th -order Runge-Kutta methods can be computed. CFL = c0
On the other hand, the dissipation and dispersion errors, introduced by the time difference scheme, may not become too large. One can impose a maximum value r = |1 − |r∗ (ω∆t)|| for the dissipation error and the same can be done for the dispersion error ω∆t = |ω ∗ ∆t − ω∆t|. The minimal value ω∆t for which both conditions are satisfied can be used as an accuracy limit L for the numerical scheme. The CFL-number can be calculated using this value L: ∆t L ≤ ∗ (20) ∆x k ∆x The accuracy limits L of the different Runge-Kutta schemes are given in table 9 for three different values of r and ω∆t . CFL = c0
r = ω∆t 0.01 0.001 0.0001
RK5 1.41 0.99 0.71
RKO5 1.66 1.38 0.41
LDDRK5 1.64 1.35 0.58
RK6 1.76 1.28 0.96
RKO6 1.90 1.47 0.55
LddRK6 1.75 1.33 0.80
Table 9: The accuracy limits L of the different Runge-Kutta schemes with different values for the maximal dispersion and dissipation errors
Based on these values the maximum CFL-numbers that are needed for time accurate numerical solutions can be calculated. All the values of table 9 are smaller than the upper limit R which was required for stability reasons. This shows that, to get time accurate solutions with a minimum of dispersion and dissipation, timesteps much smaller than the one allowed by the stability limit are necessary.
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Validation for the 1D convective wave equation
In order to validate the different numerical schemes, the initial value problem of a one-dimensional convective wave equation is considered. Dimensionless variables with length scale ∆x and time scale ∆x/c0 (with c0 the speed of sound) are used. The mathematical problem is: ∂u ∂u + =0 ∂t ∂x
(21)
with an initial disturbance u = f (x) at time t = 0. The exact solution consists of the initial disturbance propagating to the right with a dimensionless speed, equal to 1 (u = f (x − t)). Different initial disturbances can be used to evaluate some specific properties of the different numerical schemes. Three different initial disturbances will be used in this paper: • Gaussian function (eq. (22)): An initial disturbance in the form of a Gaussian function is used to evaluate the general dispersion and dissipation properties and the stability ranges of the different numerical schemes. " 2 # x f (x) = 0.5 exp − ln 2 (22) 3 The Fourier transformation of a Gaussian pulse is also a Gaussian pulse with a maximum at k∆x = 0. The initial disturbance will therefore be mainly situated in the long wave range and artificial damping is not necessary and will not be added to the numerical scheme. • ”Boxcar” function (eq. (23)): The effect of artificial damping can be evaluated with discontinuous initial conditions. The initial disturbance used for this test problem, can be written as: f (x) = H(x + M ) − H(x − M )
(23)
with H(x) the unit step function and M = 25 the half-width of the boxcar. The Fourier transformation of this function contains high frequency components and these must be damped in order to avoid spurious high-frequency waves. For this reason, this function can be used to evaluate the effect of artificial damping. • Sine function (eq. (24)): The minimal resolution of the numerical scheme can be checked with a sine function as initial disturbance. The Fourier transformation of a sine function is a dirac function, which must lie in the long wave range, so that no artificial damping must be added to the numerical schemes. f (x) = 0.5 ∗ sin(
2πx ) λ
− λ/2 ≤ x ≤ λ/2
(24)
All calculations are carried out with the above described schemes. The variables of eq.(21) are made dimensionless with ∆x as length scale and ∆x/c0 as the time scale. The computational domain has a dimensionless length of 500. For the evaluation of the numerical schemes, the result at a dimensionless time of 400 are calculated. At this time, the initial disturbance reaches the dimensionless position 400. No specific boundary conditions have been added since spurious reflections of the initial disturbance at the boundaries are negligible if the disturbance has not reached the boundary yet. For each calculation, the error will be estimated by the residual Res: the total difference between the exact solution u and the numerical approximation u∗ . Res =
N X
(uj − u∗j )2
j=1
with N the total number of grid points in the one-dimensional computational domain.
(25)
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3.1 Initial disturbance: Gaussian function A first calculation was done with a space step ∆x equal to 1 and a time step ∆t equal to 1/10. The CFLnumber of the PDE (∆t/∆x) equals 0.1, which is much smaller than the minimum CFL-numbers that could be calculated with the accuracy limits (eq. (20)). The main error will then be introduced by the spatial difference scheme. Figure 5 plots the results for the four spatial schemes with the LddRK6-scheme. For the other temporal difference schemes, the same conclusions hold.
Figure 5: Comparison between the exact (dotted line) and computed (full line) solutions of the one-dimensional wave equation at t=400 (top left=ST7+LddRK6, top right=ST9+LddRK6, bottom left=DRP+LddRK6, bottom right = STO9+LddRK6)
The numerical solutions are in good agreement with the exact solutions. But there are still some dispersed waves trailing behind the main pulse. The reason for the presence of these trailing waves can be understood from figure 2. All long waves propagate with the same group velocity and with the same propagation speed as the PDE. From a certain wavenumber k∆x onwards, the nondimensional group velocity is less than one. This means that from this frequency the wavecomponents propagate at a speed which is smaller than the propagation speed of the PDE. The spatial Fourier transformation of the Gaussian pulse contains as mainly low-frequency components which are propagated almost without dispersion. But there are still some high-frequency components that are propagating at a lower wavespeed. These components are responsible for the trailing wave behind the main pulse. The trailing waves are the most apparent for the ST7-scheme. Since the DRP-scheme is constructed to have better dispersion properties than the ST7-scheme, the trailing waves are still visible but they have a smaller amplitude. The ST9- and STO9-schemes have the best performance but are computationally more expensive due to the involved 9-point stencil. The same calculations can be made with a space step ∆x equal to 0.5 and a time step ∆t equal to 1. The CFLnumber equals 2 which is close to the stability- and accuracy limit of most of the temporal finite difference schemes. Figure 6 plots the results for five time advancing schemes with the DRP-scheme (RK6 is unstable for CFL equal to 2). For the other spatial difference schemes, the same conclusions hold. The error in the numerical solution is introduced mainly by the time integration. The space step has become two times smaller so waves are accurately propagated to a wavenumber that is twice the wave number of the first example. There are some trailing waves introduced by the RK5-scheme, the trailing waves introduced by the optimized schemes are negligible. The LddRK6 and RKO6 clearly perform the best but they demand more calculation time since they are both 6-stage methods. In figure 7 the stability range of the different numerical schemes is drawn for a constant space step ∆x = 1. The stability range is clearly extended if the temporal scheme is optimized for a minimal dispersion and
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Figure 6: Comparison between the exact (dotted line) and computed (full line) solutions of the one-dimensional wave equation at a dimensionless time t=400 (top left= DRP+RK5, middle left= DRP+RK05, bottom left = DRP+LddRK5, top right = DRP+RKO6, bottom right=DRP+LddRK6)
dissipation. If the spatial schemes are compared, the 9-point stencil (STO9) has the smallest stability range and both 7-points stencils (ST7 and DRP) become unstable at more or less the same CFL number. If one looks at the residuals of the different numerical schemes the results are as expected: the schemes based on a 9-point stencil are the most performant ones and the ST7-scheme has the largest residual. The performance of the DRP-scheme in combination with the RKO5-scheme is remarkably poor.
Figure 7: Residuals of the different schemes (∆x = 1, CFL= ∆t)(-=ST7, -.-=DRP, ...=STO9, ...=ST9)
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3.2 Initial disturbance: Boxcar The efficiency of the different damping curves must be evaluated with a discontinuous initial disturbance. The Fourier transformation of such a function also contains high-frequency components which will be propagated at a different group velocity and generate spurious waves and grid-to-grid oscillations. A ’boxcar’ function is used. In figure 8 the three different damping curves are applied to the convective wave equation with the DRP- and LddRK6-schemes. By looking at the solution without artificial damping one clearly observes a large amount of spurious waves. These waves are present everywhere in the computational domain. All the damping curves eliminate a large part of the spurious oscillations.
Figure 8: Propagation of boxcar function with different damping curves with damping constant µ = 0.3 at a dimensionless time t=400 (dotted line = exact solution, solid line = numerical solution)(∆x = 1,∆t=1) When a Gaussian function is used as a damping function (D1, D2), the efficiency of the damping is much better for a function with a larger half-width (D1). The use of the Gaussian-like function (D3) with a very large half-width damps out the spurious oscillations but generates a large overshoot at the discontinuities of the disturbance. When a small half-width is used (D2) there are still some wave-components that are not sufficiently damped out. These components generate waves with a slightly different propagating speed so they are only present in the vicinity of the discontinuity. Another parameter of the artificial damping is the value of the damping constant µ (eq.(8)). Three different values are used: µ = 0.1, 0.3 and 0.5 (Fig. 9). When a larger value of µ is used, there is more damping added in the scheme and the amount of spurious waves becomes smaller. From µ = 0.3 onwards, not much difference is obtained by enlarging the value of the damping constant. When the damping constant becomes too large (µ > 0.7) the numerical scheme becomes unstable and no solution can be obtained.
3.3
Initial disturbance:Sine function
Finally the minimal resolution (points-per-wavelength) of the different numerical schemes is considered. When the resolution is too small the Fourier transformation of the sine function will contain parts that are not resolved very well by the numerical scheme and the sine wave will be propagated together with a number of spurious waves. Starting from a minimal resolution the solution will contain less spurious waves and the
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Figure 9: Propagation of boxcar function with different values of the damping constant µ calculated with damping function D2 at a dimensionless time t=400 (dotted line = exact solution, solid line = numerical solution)(∆x = 1,∆t=1) sine function can be clearly separated from these trailing waves. This number is the minimal number of grid-points per wavelength required for the numerical scheme. The result for three different resolutions with the DRP- and LddRK6-scheme are plotted in figure 10.
Figure 10: Propagation of sine function with different resolution at a dimensionless time t=400 for the DRP- + LddRK-scheme (∆x = 1,∆t=1) (dotted line = exact solution, solid line = numerical solution)
In figure 11 the residuals of the different numerical schemes are shown for a number of different resolutions. The minimal resolution (the resolution at which the sine function is propagated accurately enough) is ob-
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tained when the residual is approximately less than 0.15. The minimal resolution is much larger than the values given in table 8 due to the fact that there are also dispersion and dissipation errors generated by the time advancing schemes. But the same trends as in Table 8 can be observed: STO9 has the smallest minimal resolution, ST5 the largest. When the temporal finite difference scheme is optimized, the minimal resolution becomes smaller. One must remark that the classical rule of thumb of 8 to 9 grid points per wavelength is only valid for some numerical schemes. A minimum of 12 to 14 grid points per wavelength is a much better approximation.
Figure 11: Residuals of the propagation of a sine function with different resolution (points-per wavelength) for the different numerical schemes (∆x = 1,∆t=1) (red: -=RK5, —=RKO5, ...=LddRK5, blue: -=RK6, —=RKO6, ...=LddRK6)
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Conclusions
A number of numerical finite difference schemes specially suited for CAA-computations have been reviewed in this paper. For space integration, the standard 6th - and 8th -order central difference schemes, the Dispersion-Relation-Preserving scheme and the optimized 9-point central difference scheme are investigated. The DRP- and the STO9-scheme have better dispersion properties than the ST7- or ST9-scheme. The STO9-scheme is clearly the superior scheme, but requires more calculation time than the 7-point DRPscheme. Artificial damping is required to avoid spurious short waves. A Gaussian function with a broad half-width and a damping constant of 0.3 is a compromise between avoiding spurious oscilations and minimizing the damping of the right solution. Different kinds of Runge-Kutta schemes are used as time advancing schemes: standard 5-stage and 6-stage Runge-Kutta schemes, optimized 5- and 6-stage Runge-Kutta schemes and the 5- and 6-stage low dispersion, low dissipation Runge-Kutta schemes. The 6-stage optimized schemes (RKO6, LddRK6) have the best dispersion and dissipation properties but require more calculation time. The RKO5 and LddRK5 schemes are also optimized for dispersion and dissipation properties. Together with their 6-stage variants, the optimized RK-schemes exhibit a larger stability comparing to the traditional RK-schemes (RK5, RK6).
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In this paper, the four different space difference methods are combined with the six Runge-Kutta methods. The combination of the two methods makes it possible to make some important conclusions for the different numerical methods: • The choice of the minimal timestep is limited by the accuracy range and not by the stability range. • To get accurate results a minimal spatial resolution of 12 to 14 grid points per wavelength is required. The combination of the optimized schemes has properties that are ideal for CAA applications: • A low dissipation and dispersion error makes the numerical scheme suited for waves propagating over large distances. • A large stability range (especially for the DRP- + LddRK-schemes) reduces the calculation time since larger time steps can be used. • Both optimized schemes need less grid-points per wavelength than the classical methods. A numerical scheme for CAA-applications must be a compromise between minimal computational efforts and a minimal dispersion and dissipation error. The combination of the DRP- with the LddRK5-scheme or of the STO9- with the RKO5-scheme seem the best methods that meet these properties.
Acknowledgements The research work of Wim De Roeck is financed by a scholarship of the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT).
References [1] C.K.W. Tam, Computational Aeroacoustics: Issues and Methods”, AIAA-Journal, Vol. 33, No. 10, p. 1788-1796 (1995). [2] J.W. Goodrich, A Comparion of Numerical Methods for Computational Aeroacoustics”, Proceedings of the 5th AIAA/CEAS Aeroacoustics Conference, Bellevue, WA, USA (1995). [3] C.K.W. Tam, J.C. Webb Dispersion-Relation-Preserving Schemes for Computational Aeroacoustics”, Proceedings of the 14th DGLR/AIAA Aeroacoustics Conference, Aachen, Germany (1992). [4] C. Bogey, C. Bailly A Family of Low Dispersive and Low Dissipative Explicit Schemes for Computing the Aerodynamic Noise”, AIAA-paper 2002-2509, (2002). [5] F.Q. Hu, M.Y. Hussaini, J.L. Manthey Low-Dissipation an Low-Dispersion Runge-Kutta Schemes for Computational Aeroacoustics”, Journal of Computational Physics, No. 124, p. 177-191 (1996). [6] C.K.W. Tam, J.C. Webb, Z. Dong A Study of the Short Wave Components in Computational Acoustics”, Journal of Computational Acoustics, Vol.1, No. 1, p. 1-30 (1993).
