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*Corresponding Author: Fazal Ghaffar, Department of Mathematics, Abdul Wali Khan University. Mardan, KPK, Pakistan.
Email:[email protected]
Multigrid Method for 2D Helmholtz Equation using Higher Order Finite Difference Scheme with variable wave number(k)
Fazal Ghaffar1*, Noor Badshah2, Murad Ullah3, Muhammad Altaf Khan1, S. Islam1
1Department of Mathematics, Abdul Wali Khan University. Mardan, KPK, Pakistan.
2Department of Basic Sciences, UET, Peshawar Pakistan.
3Department of Mathematics, Islamia College University, Peshawar, KPK, Pakistan.
Received: June 19, 2014 Accepted: September 22, 2014 ABSTRACT
A sixth-order compact difference scheme with variable wave number k and uniform mesh sizes in different coordinate directions is employed to discretize a two dimensional Helmholtz equation. Numerical experiments are conducted to test the accuracy of the Sixth-order compact difference scheme with multigrid method and to compare it with the fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme is established by using the errors norms L2.
KEYWORDS: Helmholtz equation, Sixth-order compact scheme, Multigrid method, Krylov subspace.
1 INTRODUCTION
The struggle for computing accurate solution using different grid sizes have increased researchers curiosity for developing high order difference schemes. Compact finite difference scheme is widely used in vast area of computational problems such as, the Helmholtz equations and other elliptic equations [7,1]. We seek high-accuracy numerical solution of the two-dimensional Helmholtz equation as
uxx+u
yy+k2(x)u=f(x,y),(x,y)∈Ω. (1)
where Ω is a rectangular domain. The above equation is an elliptic partial differential equation (PDE). The above equation has broad application in physical phenomena, such as elasticity, electromagnetic waves, potential in time harmonic acoustic and electromagnetic fields, acoustic wave scattering, water wave propagation, noise reduction in silencers, radar scattering and membrane vibration are governed in frequency domain. The solution u(x,y) and the forcing function f(x,y) are assumed to have the required continuous partial derivatives up to specific orders and sufficiently smooth and . In this study we analyze finite difference approximation on uniform grids ∆x=∆y in both x and y- directions. In this study we use a variable values of k to obtain a scheme with sixth order of accuracy [8]. The second-order central-difference operators defined at (xi,yj) can be written as:
,
, (2)
equation (1) can be discretized at a given grid points (xi,yj) as :
(3)
where O(h2
) denotes the truncated terms on the order O(h2 1+h2
2). Helmholtz equation has been numerically solved by different techniques and different approaches are developed such as the finite-difference method [6], the finite- element method [3], the spectral-element method [4], compact finite-difference method [5].
In finite-difference methods, the stencil of grid points needs to be enlarged, in order to increase the order of accuracy of approximation but this is not desirable. Helmholtz equation is extensively solved by finite-element methods, but the disadvantage of finite-element method is the high computational cost, another disadvantage is the pollution effect i.e its results are less accurate solution at higher wave number for the given nodes per wavelength. In spectral-element method, it shows that, it requires fever grid nodes per wavelength as compared to the finite-element methods for the Helmholtz equation [4]. But due to the less sparse resultant matrix compared to the resulting finite- element matrix, computational time of both methods are the same [4]. Many iterative techniques for the Helmholtz
equation supper due to their slow convergence, when high frequencies need to be determined. The struggle for fast iterative methods for high-frequency Helmholtz equations become the center of active research.
In general to obtain more accurate numerical solution by adding more nodes and using smaller mesh sizes, which requires more computational time and storage space. A noticeable work in this field has been done by Turkel and Singer [6]. They developed a fourth-order compact finite-difference method using Dirichlet and/or Neumann boundary conditions. After this M.H.K siddique [5] developed a sixth-order finite-difference method for solving Helmholtz equation in one-dimensional and two-dimensional domain with constant wave number k. In this work the basic issue discussed is to develop the multigrid method to solve Helmholtz equation using special type of smoothers with variable wave number k. In order to accelerate multigrid method we are using Krylov subspace. However multigrid method alone is also convergent rapidly, but in some cases it converges slowly. The present study is the first that using sixth- order compact finite-difference scheme with variable wave number k and the designing of specialized multigrid method also using Krylov subspace as an accelerated multigrid.
2 The Model Problem
Consider the two-dimensional Helmholtz Equation
uxx(x,y)+uyy(x,y)+k2(x)u(x,y)=f(x,y),(x,y)∈Ω. (4)
In order to achieve an appropriate description, a Taylor series expansion is performed for the descritized field where the grid spacing ∆x=∆y =h.
ui+1,j=u i,j+h∂
xu i,j+ h2
2∂2 xu
i,j+ h3 6∂3
xu i,j+ h4
24∂4 xu
i,j+ h5 120∂5
xu i,j+ h6
720∂6 xu
i,j+O(h7), (5)
ui−1,j=ui,j−h∂xui,j+ h2 2∂2
xui,j− h3 6∂3
xui,j+ h4 24∂4
xui,j− h5 120∂5
xui,j+ h6 720∂6
xui,j+O(h7
), (6)
Adding the above expressions and solving for the second derivative which gives :
(7) Using equation (2) and (7), we have
δ2 xu
i,j=∂2 xu
i,j+ h2 12∂4
xu i,j+ h4
360∂6 xu
i,j+O(h6), (8)
Similarly we can find the approximation for the variable y. Therefore the central difference scheme for Helmholtz equation can be written as :
δ2 xui,j+δ2
yui,j+k2
(x)(ui,j)+αi,j=fi,j+O(h6
), (9)
where
αi,j=− h2 12
∂4ui,j
∂x4 +
∂4ui,j
∂y4 i,j
− h4 360
∂6ui,j
∂x6 +
∂6ui,j
∂y6 i,j +O(h6
). (10)
The higher order derivatives is expressed through mixed derivatives by successively differentiating equation (4), this process also includes derivative of the forcing function f(x,y). Applying the appropriate derivatives of equation (4), we get
∂4u
∂x4 i,j
=
∂2f
∂x2−k2 ∂2u
∂x2− ∂4u
∂x2∂y2 i,j ,
∂4u
∂y4 i,j
=
∂2f
∂y2−k2 ∂2u
∂y2− ∂4u
∂y2∂x2 i,j
(11) putting above equation in equation (10), we get
αi,j=− h2 12
∂2f
∂x2+ ∂2f
∂y2 i,j
−k2
∂2u
∂x2+ ∂2u
∂y2 i,j
−2
∂4u
∂x2∂y2 i,j
− h4 360
∂6u
∂x6+ ∂6u
∂y6 i,j
+O(h6). (12)
Now in order to find the fourth-order approximation of
∂4u
∂x2∂y2 i,j
in above equation, which can be obtained from tailer series expansion such that
∂4u
∂2x∂2y i,j
=δ2 xδ2
yui,j− h2 12
∂6u
∂x4∂y2+ ∂6u
∂x2∂y4 i,j +O(h4
), (13)
Now putting equation (13) in equation (12), we get αi,j= h2
12
−∇2f
i,j+2δ2 xδ2
yu i,j+k2f
i,j−k4ui,j - h4 360
∂6u
∂x6+ ∂6u
∂x6+5 ∂6u
∂x4∂y2+5 ∂6u
∂x2∂y4 i,j
+O(h6). (14)
For compact sixth-order approximation we need expression of the four derivatives of order sixth in equation (14), Differentiating equation (4), we have
∂6u
∂x4∂y2+ ∂6u
∂x2∂y4+k2 ∂4u
∂x2∂y2 i,j
=
∂4fi,j
∂x2∂y2, (15)
∂6u
∂x6+ ∂6u
∂y6 i,j
=∇4 fi,j−k2
∂4u
∂x4+ ∂4u
∂y4 i,j
−
∂6u
∂x4∂y2+ ∂6u
∂x2∂y4 i,j
, (16)
putting equation (11) and equation (15) in equation (16), we get
∂6u
∂x6+ ∂6u
∂y6 i,j
=∇4 fi,j−
∂4f
∂x2∂y2 i,j +3k2
∂4u
∂x2∂y2−k2∇2f+k2(−k2u+f) i,j
, (17)
Using equation (15) and equation (17), also using all different derivative values of ui,j used in equation (14), that is α
i,j = h2 12(−∇2
fi,j+2δ2 xδ2
yu i,j+k2
fi,j−k4ui,j) −
h4 360(∇4f
i,j−k2∇2fi,j+4
∂4f
∂x2∂y2 i,j
−2k2δ2 xδ2
yu
i,j−k6ui,j+k4f
i,j)+O(h6). (18)
Consequently, the compact sixth-order approximation of the two-dimensional Helmholtz equation be written in modified form which leads to:
h2 6
1+ k2 h2 30 (δ2
xδ2 y)v
i,j+(δ2 x+δ2
y)v i,j+k2
1− k2 h2 12 + k4
h4 360 v
i,j=
1− k2 h2 12 + k4
h4 360 fi,j+
h2 12(1− k2
h2 30 ) ∇2
fi,j+ h4 360∇4
fi,j+ h4 90
∂4f
∂x2∂y2 i,j
, (19)
vi,j is the approximation to u
i,j which satisfies the original equation (14), and v i,j=u
i,j+O(h6) . (δ2
xδ2 y)vi,j= 1
h4[(vi+1,j+1+vi+1,j−1+vi−1,j+1+vi−1,j−1)−2(vi,j+1+vi,j−1+vi+1,j+vi−1,j)
+4(vi,j)], (20)
using equation (2) and equation (20), the left hand side of equation (19) can be written as : h2
6
1+ k2 h2 30 (δ2
xδ2
y)vi,j+(δ2 x+δ2
y)vi,j+k2
1− k2 h2 12 + k4
h4
360 vi,j=a1vi,j+a2A11+a3A12, (21) where
a1=− 10 3+k2
h2
46 45− k2h2
12 + k4h4
360 , , a3= 1 6+ k2h2
180, and
A11=vi+1,j+vi−1,j+vi,j+1+vi,j−1,
A12=vi+1,j+1+vi+1,j−1+vi−1,j+1+vi−1,j−1.
Also the right hand side of equation (19) can be express as:
b11f i,j+b
12∇2f i,j+b
13∇4f i,j+b
14
∂4f
∂x2∂y2 i,j
, (22)
where ∇2 is the laplace operator and ∇4 is the bi-harmonic operator. The above equation can be written in most general form as :
b1f i,j+b
2B 11+b
3B
12, (23)
where b1= h2
90
63− 13h2 k2 2 + h4
k4 4 , b
2= h2 360(23−h2
k2
), b3= h2 90, and
B11=f i+1,j+f
i−1,j+f i,j+1+f
i,j−1, B12=f
i+1,j+1+f
i+1,j−1+f
i−1,j+1+f i−1,j−1.
Thus the compact sixth-order approximation of two-dimensional Helmholtz equation can be written in more general form
a1vi,j+a2A11+a3A12=b1fi,j+b2B11+b3B12. (24) In equation (24), we assume that the derivative of the right hand side function f
i,j can be determined numerically. In case where f
i,j is not known analytically, then one can approximate the derivatives with high order finite differences.
we need only a 4th-order accurate approximation of ∇2f
i,j and a 2nd-order accurate approximation of ∇4f i,j and
∂4f
∂x2∂y2 i,j
. Fourth-order accurate approximation to ∇2f
i,j is obtained by using (1− h2
12δ2)−1 δ2fi,j. Which can be further express
1
12h2[−fi+1,j−fi−1,j+16f i+ 1
2,j+16f i− 1
2,j−60fi,j+16f i,j+ 1
2
+16fi,j− 1 2
−fi,j+1−fi,j−1]. (25)
Similarly the second-order accurate approximation to ∇4f
i,j can be express as
∇4
fi,j=[ ∂4f
∂x4+ ∂4f
∂y4]= 1 h2[f''''
i+1,j+f'''' i−1,j+f''''
i+1,j+f'''' i−1,j−4f''''
i,j]. (26)
And the second-order accurate approximation of [ ∂4f
∂x2∂y2] is [ ∂4f
∂x2∂y2]=(δ2 xδ2
y)fj,k= 1 h2[f''
i+1,j+f'' i−1,j+f''
i+1,j+f'' i−1,j−4f''
i,j]. (27)
Using the usual natural ordering on equation (19), leads to a system of equation of the form Au=b, where A is a sparse and symmetric matrix.
3 Multigrid Method
The results of fourth-order compact difference and sixth-order accurate finite-difference scheme are in sparse linear systems, can be solved efficiently by multigrid methods. In order to remove high frequency error, multigrid method utilizes some relaxation methods. Multigrid method makes the use of coarse grid correction to remove smooth errors. Efficient multigrid method are implemented in [2, 10] for solving 2D poisson equation discretized by the standard fourth-order compact difference schemes. It is shown that the sixth-order compact difference scheme is most cost effective than the fourth-order compact difference schemes and the corresponding conventional second-order central-difference scheme with multigrid methods. The standard multigrid method with a point Guass-Seidel type relaxation and standard mesh coarsening strategy does not work very well with unequal mesh sizes discretized poisson equation [9].
The strategy used is the line relaxation to replace point relaxation. That is the line Guass-Seidel relaxation can be shown to be very effective in removing high frequency errors. The coefficient matrix of the sixth-order compact finite difference with this ordering can be written in block tri-diagonal matrix of the block order (N−1). The order of the coefficient matrix U is of (N−1)×(N−1), where U=diag[U
1,U 0,U
1], where U
0=diag[a 2,a
1,a 2], U
1=diag[a 3,a
2,a 3], are both symmetric tri-diagonal sub-matrices of order (N−1). They represent the sub matrix of each grid line along one direction. Where u
j is the part of the solution vector representing the grid points on each jth line, and f j is the corresponding part on the right-hand side vector. On each level U0 needs only one factorization. The factorization cost of U0 is negligible because matrix B has constant blocks which does not change from one grid line to another grid line.
In multigrid method with LU-decomposition or Gauss-Seidel relaxations, we use bilinear interpolation to transfer correction from a coarse grid to a fine grid, and we also use a full-weighting scheme update residual on a coarse grid.
Multigrid Algorithm:
Algorithm 1 Assuming that we set up these multigrid parameters:
v1 pre smoothing steps on each level.
v2 post smoothing steps on each level.
r is the number of multigrid cycles on each level (r=1 for V-cycling and r=2 for W-cycling), here we use V-cycle with r=1.
FAS Multigrid Cycle
φh←FASCYCh(φh,fh,v1,v2,r) 1. if Ωh
is the coarsest grid, the solve equation (15) using a time marching technique and then stop.
Else do the pre-smoothing step:
φh⟵Smoother(φh,fh,v1,tol), (Pre−Smoothig) 2. Restriction:
φ2h=I2h
h φh,φ2h=φ2h, f2h=I2h
h (fh−Nhφh)+N2hφ2h, φ2h⟵FASCYC2h(φ2h,f2h,v1,v
2).
3. Interpolation:
φh⟵φh+Ih
2h(φ2h−φ2h).
4.
φh⟵Smootherv2(φh,fh,v2). (Post−Smoothing) Here the restriction operator I2h
h is by full weighting and the interpolation Ih
2h by bilinear operator.
4 Krylov subspace as an accelerated multigrid
Krylov subspace iteration methods are an important techniques based on projection used to solve large linear systems. Krylov subspace method is that method in which the subspace Km is spanned by the vectors of the krylov sequence
Km(A,r
0)=span{r−0,Ar 0,A2
r0,...Am−1 r0},
where Km is the m-th Krylov subspace and r0=b−Ax0. The standard multigrid method is modified in the following manner to solve helmholtz equation.
1. We use krylov subspace iteration such as Generalized Minimum Residual Method(GMRES)as a smoother which provide a maximum reduction of oscillatory components.
2. We use multigrid method as a preconditioned for a GMRES iteration for the original equation Ax=b, to handle modes with eigen values that are closed to the origin on all grids.
We use V-cycle multigrid method and GMRES iteration preconditioned by the same V-cycle multigrid method. The outer iteration is run until the stopping criterion
|rm|
|r0|<10−6 is satisfied. Where rm is the residual of m-th iteration.
5 Numerical Calculations
We consider two dimensional problem with known analytic solution. In example three parameters a, b and c with same variable k is applied such as: k(x)=a−bsin(x) with a>b≥0.
Thus a , b control the range of k while |c| controls the number of oscillations of k in a specific domain. We will discuss two cases for c=10 and a=10 in both, so k has five periods as in fig1.
Case 1: b=9 (1≤k≤19, 3.14≤k≤59.65 ).
Case 2: b=4 (6≤k≤14, 18.85≤k≤43.98 ).
In order to show the efficiency and applicability of the multigrid method, numerical experiments are conducted to solve a 2D Helmholtz equation (1) on the unit square domain D=π×π.
Example 1 uxx(x,y)+u
yy(x,y)+k2(x)u(x,y)=−b(2a+c)sin(cx)e−k(x)/csin(y). (28)
Figure 1: Plot of k(x) for a=10, b=9, c=10.
The sixth-order compact finite difference scheme is compared against the standard second-order central difference scheme and fourth-order compact difference scheme, in terms of solution of accuracy, multigid convergence rate, and CPU timing. All multigrid methods use V-cycle or W-cycle algorithm, the coarsest grid is the one with at least one dimension being the coarsest possible. One pre-smoothing and one post smoothing are applied at each level. The iteration stop when the Euclidean norm (2-norm) of the residual vector is reduced by 10−12. The maximum absolute error reported is the maximum absolute error between the computed solution and the exact solution over the entire fine grid points. In order to compare the numerical solution and the exact solution, we use l
2-norm . The matrix l
2-norm of the error vector is defined as:
|e2|= 1
N
i,j=0 N
e2
i,j , (29)
where the error vector e ij=u
ij−vij and the residual r=(f−Av)=Ae, N is the number of nodes, k(x) is the wave number, e1
=101
also M2 is MG with 4th-order, M3 is MG with 6th-order and M4 is MG with sixth order accelerated by krylov subspace. The l2-norm of the error for and different values of N and k, are presented in the following table.
Table 1: Comparison of Maximum Absolute errors and CPU(Seconds) for a Multigrid Method with Difference Discretization Schemes, for example 1, |e
2|, a=10, b=9, N=4,8,16,32,64,128
N M
2(|e
2|) CPU(Second) V-Cycles M
3(|e
2|) CPU(Second) V-Cycles
4 2.2284e−4 0.042 2
6.2252e−7 0.042 2
8 9.7200e−5 0.055 2
3.2845e−7 0.054 2
16 1.9170e−6 0.061 2
4.5025e−9 0.060 2
32 6.9331e−8 0.088 2
6.6001e−11 0.087 2
64 3.5772e−9 0.322 2
9.9922e−13 0.313 2
128 2.1063e−10 2.233 2
1.5369e−14 2.224 2
Table 2: Comparison of Maximum Absolute errors and CPU(Seconds) for a Multigrid Method with Difference Discretization Schemes, for example 1, |e
2|, a=10, b=4, N=4,8,16,32,64,128
N M
2(|e
2|) CPU(Second) V-Cycles M
3(|e
2|) CPU(Second) V-Cycles
4 3.8792e−4 0.042 2
8.4255e−7 0.042 2
8 9.9621e−5 0.055 2
5.8086e−7 0.054 2
16 3.9761e−6 0.061 2
6.6055e−9 0.060 2
32 7.3493e−8 0.088 2
8.3216e−11 0.087 2
64 3.7782e−9 0.322 2
9.9988e−13 0.313 2
128 3.1431e−10 2.233 2
2.8763e−14 2.224 2
We have examine the behavior of scheme for different values of k. Error is reducing versus k increasing. We also observed that using Krylov subspace as a smoother the error is reduced and the residuals decreases versus the number of restarts. We have also examine the behavior of scheme for different values of N.
The data in table 1, 2 indicates the behavior of N and k for different values. The data in table 3, 4 and 5 indicates the behavior of error and residuals where Krylov subspace is applied. In figures 2 the left figure is the error graph of both problems. The restarted GMRES convergence rate is indicated in figure 3.
Example 2 uxx(x,y)+u
yy(x,y)+k2u(x,y)=−2[(1−6x2)(y2−y4)+(1−6y2)(x2−x4)]
∈ [ ]
. (30)With the pure Dirichlet boundary conditions on all side of a unit square, i.e u(0,y)=u(1,y)=u(x,0)=u(x,1)=0,
The exact solution for example 2 is u(x,y)=(x2−x4)(y4−y2) .
Table 3: Comparison of Maximum Absolute errors and CPU(Seconds) for a Multigrid Method with Difference Discretization Schemes for example 2, |e
2|, k=100, N=4,8,16,32,64,128
N M
1(|e
2|) CPU V M
2(|e
2|) CPU V M
3(|e
2|) CPU V
4 6.0562e−2 0.062 2
6.3220e−6 0.042 2
5.6567e−7 0.042 2
8 1.2872e−4 0.074 2
8.1311e−7 0.059 2
7.5902e−8 0.057 2
16 3.2304e−5 0.080 2
3.2134e−7 0.080 2
6.2834e−8 0.063 2
32 8.1531e−6 0.093 2
9.8314e−8 0.187 2
5.5698e−8 0.181 2
64 2.0389e−6 0.118 2
7.0389e−8 0.329 2
5.2102e−8 0.313 2
128 5.0970e−7 0.290 2
6.9061e−8 2.269 2
5.1442e−8 2.252 2
Table 4: Comparison of Maximum Absolute errors ,Convergence rate and Residuals for a Multigrid Method with Krylov subspace, for example 1,|e2|, a=10, b=9, N=4,8,16,32,64,128
N M
3(|e
2|) Res Conv-
rate
M4(|e
2|) Res Conv-
rate
4 6.2252e−7
1.6889e−13 7.0856
2.2284e−7
2.7661e−16 6.9530
8 3.2845e−7
3.1793e−14 7.0856
5.5413e−8
4.5025e−15 9.1802
16 4.5025e−9
4.2357e−14 9.2496
3.3681e−9
4.9474e−15 11.3443
32 6.6001e−11
6.5354e−14 11.3840
3.6209e−11
4.5025e−9 13.4792
64 9.9922e−13
1.3362e−13 13.5055
5.3078e−13
1.0692e−14 15.6001
128 1.5369e−14
2.7153e−13 15.6195
1.2354e−15
2.0906e−14 17.7144
Table 5: Comparison of Maximum Absolute errors, Residuals and Convergence rate for a Multigrid Method with Klylov subspace, for example 2, |e
2|, k=100, N=4,8,16,32,64,128
N M
3(|e
2|) Res Conv-
rate
M4(|e
2|) Res Conv-
rate
4 5.6567e−7
7.3562e−7 2.4223
3.2845e−7
2.2346e−7 3.2545
8 7.5902e−8
3.7529e−7 3.0570
6.2834e−8
2.6743e−7 3.5201
16 6.2834e−8
5.6561e−7 3.0632
5.6437e−8
4.5321e−7 3.5431
32 5.5698e−8
5.8652e−7 3.1810
4.9156e−8
4.2264e−5 4.2422
64 5.2102e−8
2.6240e−2 3.8313
2.2351e−8
2.2854e−5 4.2872
128 5.1442e−8
2.7584e−2 3.9003
8.9830e−9
5.6567e−7 4.7341
Figure 2: Left: shows the error graph for example 1. Right: shows the error graph for example 2. The error vector eij=uij−vij and N=128 are the number of nodes.
Figure 2: Left: shows the error graph for example 1. Right: shows the error graph for example 2. The error vector
eij=u
ij−vij and N=128 are the number of nodes.
Figure 3: Left: when m=128×128, the restarted GMRES converges in 2.1 restarts for example 1. Right: when m=128×128, the restarted GMRES converges in 2 restarts for example 2.
Figure 4: Left: when m=128×128, the restarted GMRES converges in 2.1 restarts for example 1. Right: when m=128×128, the restarted GMRES converges in 2 restarts for example 2.
6 Conclusion
We have studied a sixth-order compact finite difference scheme with equal mesh sizes for discretizing a 2D Helmholtz equation. We have developed special multigrid methods to solve the resulting sparse system efficiently. It is observed that multigrid method with GMRES as a smoother work very well in solving the Sixth-order compact finite difference scheme-discretized 2D Helmholtz equation. We have also studied the accuracy of the scheme when the variable wave number k itself is highly oscillatory. It was found that the accuracy and convergence were equally good even when there was a large ratio between k max and k min. The pollution effect states that as the wave number k increases, the required number of subintervals (per domain side), N, to obtain a given accuracy increases according to the relation
, where p is the order of accuracy of the scheme. Numerical results shows that multigrid method accelerated by krylove subspace on Sixth-order compact scheme has the expected accuracy and more faster than fourth-order finite-difference scheme. It is also obvious from the results that the overall error decreases with an increase in the wave number.
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