Research ArticleModified Finite Difference Schemes on Uniform Grids forSimulations of the Helmholtz Equation at Any Wave Number
Hafiz Abdul Wajid1 Naseer Ahmed2 Hifza Iqbal3 and Muhammad Sarmad Arshad4
1 Department of Mathematics COMSATS Institute of Information Technology Lahore 54000 Pakistan2Department of Mechanical Engineering Faculty of Engineering CECOS University of IT and Emerging SciencesPeshawar 25000 Pakistan
3Department of Mathematics and Statistics The University of Lahore Lahore 54000 Pakistan4Department of Mathematics University College of Engineering Sciences and Technology Lahore Leads UniversityLahore 54000 Pakistan
Correspondence should be addressed to Hafiz Abdul Wajid hawajidciitlahoreedupk
Received 11 April 2014 Accepted 20 June 2014 Published 12 August 2014
Academic Editor Song Cen
Copyright copy 2014 Hafiz Abdul Wajid et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We construct modified forward backward and central finite difference schemes specifically for the Helmholtz equation by usingthe Bloch wave property All of these modified finite difference approximations provide exact solutions at the nodes of the uniformgrid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions forwave propagation The most important feature of the modified schemes is that they work for large as well as low wave numberswithout the common requirement of a very fine mesh size The superiority of the modified finite difference schemes is illustratedwith the help of numerical examples by making a comparison with standard finite difference schemes
1 Introduction
In the present era the phenomenon of waves propagating isan unavoidable part of human life as it widely covers theirdomestic social commercial security and defence purposeneeds Consequently the accurate propagation of waves hasbeen and more importantly is one of the prime concernsfor scientists engineers physicists and mathematiciansInterestingly many of the physical problems are governed bythe time harmonic form of the wave equation which is alsowell known as the Helmholtz equation [1 2] A few examplesare the propagation ofwaterwaves in coastal regions the scat-tering of waves from an elastic body and the highly commonpropagation of sound waves underwater (SONAR) It is notsurprising that for the above mentioned problems the ana-lytical methods mostly fail to provide us with a solution andhence the demand for efficient and reliable numerical meth-ods to solve the Helmholtz equation is obvious Much efforthas been invested in this regard and today we have a range of
numerical methods for solving the Helmholtz equation forinstance the finite volumemethod [3] finite elementmethod[4 5] finite difference method [6ndash10] and spectral elementmethod [11ndash13] which are commonly used to simulate waveswith both time independent and dependent natures
The only challenge linked with the cost of numericalsimulations for the Helmholtz equation (or in general forwave propagation problems) is the rule of thumb [2 14 15]which demands at least ten elements per wavelength forthe accurate propagation of waves (meaning dispersion anddissipation free propagation) This constraint is a seriousworry for the wave propagation community because for apractical application the physical domain can have a lengththat is thousands of times the wavelength of the wave Moreimportantly this constraint becomes a real challengewhenweare interested in the propagation of waves with a very highwave number as a high wave number means (more oscilla-tions) a smaller wavelength and for this case the amount ofcomputer resources such as memory and CPU time required
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 673106 9 pageshttpdxdoiorg1011552014673106
2 Journal of Applied Mathematics
by conventional (or standard) schemes is prohibitivelylarge
Standard finite difference schemes have extensively beenused [6ndash10] with the intention of having a precise answerto the major question ldquoHow can dispersion be eliminated orreducedwith less computational expenserdquo for the simulationof wave propagation However a serious impediment ofthe standard schemes developed so far is that they mayproduce optimal results for wave propagation problems (119886)for low wave numbers or (119887) with very high computationalcost but completely fail for high wave number propagationsHence the motivation is obvious for the development ofdiscretization schemes which are less expensive and highlyaccurate and efficient for the propagation of waves with veryhigh wave numbers
In this regard Nehrbass et al [8] in 1998 gave anew dimension to the second-order central finite difference(CFD) scheme by redefining the usual second-order CFDapproximation with a central node of 2 cos(119896ℎ)+ℎ21198962 insteadof 2Themajor limitation of this work was that (119886) it requiredprior information about the general solution of theHelmholtzequation and (119887) it did not provide exact results for problemswith nonreflecting boundary conditions However the fasterconvergence of this scheme was of no comparison to thestandard CFD scheme especially when the problem is solvedwith Dirichlet boundary conditions Recently in 2010 theidea of Nehrbass et al was further taken up by Wong andLi in [10] They proposed a different approach to solve theHelmholtz equation and their construction did not requirethe use of the general solution but instead they used theHelmholtz equation itself recursively In addition to thatthey also made the first derivative present in the radiationboundary condition exactThemost important feature of thisnew scheme was that it works for low as well as for very highwave numbers without the common weakness of fine meshsize even with radiation boundary conditions
The salient features of both schemes are (119886) easy (cheap)implementation which only requires the replacement of thecentral node (119887) the sparse tridiagonal structure of thematrixis preserved as one has in the case of the standard second-order CFD scheme (119888) both schemes provide nodally exactvalues on the interior grid points for the Helmholtz equationat any wave number
In this paper we present a modified approach to havenodally exact approximations of first- and second-orderderivatives for the Helmholtz equation on uniform grids byusing the Bloch wave property This will give us dispersionfree results in one dimension and optimal results in higherdimensions With the help of this alternate approach onecan make finite difference (central as well as forward andbackward) approximations of any order exact
The organization of the paper is as follows In Sections2 and 3 modified schemes are presented for the Helmholtzequation and radiation boundary conditions respectivelyIn Section 4 the easy (cheap) implementation of modifiedschemes is discussed In Section 6 modified schemes arepresented for two-dimensional problems Sections 5 and 6are devoted to numerical examples for both one- and two-dimensional problems respectively
In order to motivate the ideas we consider the one-dimen-sional Helmholtz equation [2]
11990610158401015840(119909) + 119896
2119906 (119909) = 0 for 119909 isin R (1)
where 119896 isin C is the wave number
21 Modified Central Finite Difference (CFD) Schemes Thestandard second- and fourth-order central finite differenceapproximations of the second-order derivative at the node119909119895= 119895ℎ of the uniformly spaced grid ℎZ are as follows [6 16]
11990610158401015840
119895asymp119906119895minus1
minus 2119906119895+ 119906119895+1
ℎ2
11990610158401015840
119895asympminus119906119895minus2
+ 16119906119895minus1
minus 30119906119895+ 16119906
119895+1minus 119906119895+2
12ℎ2
(2)
with ℎ gt 0 being the distance between two adjacent nodes ofthe grid ℎZWith these approximations for (1) we obtain thefollowing stencils
119906119895minus1
+ ((119896ℎ)2minus 2) 119906
119895+ 119906119895+1
= 0 (3)
119906119895minus2
minus 16119906119895minus1
+ (30 minus 12(119896ℎ)2) 119906119895minus 16119906
119895+1+ 119906119895+2
= 0 (4)
The above finite difference approximations (3) and (4) of theHelmholtz equation admit nontrivial solutions of the form119906119895= 119890119894119895ℎ which satisfy the property
119906119895+119899
= 119890119894ℎ119899
119906119895
forall119899 isin Z (5)
which is known as the discrete Bloch wave property [4]Moreover is the discrete wave number provided that itsatisfies
ℎ = cosminus1 (1 minus (119896ℎ)2
2)
ℎ = cosminus1 (4 minus radic9 minus 3(119896ℎ)2)
(6)
or
ℎ minus 119896ℎ = +(119896ℎ)3
24+ sdot sdot sdot
ℎ minus 119896ℎ = +(119896ℎ)5
180+ sdot sdot sdot
(7)
which we get upon writing the above expressions for ℎ as aseries in 119896ℎ
It is evident from (7) that second- and fourth-orderapproximations of the second derivative present in theHelmholtz equation provide second- and fourth-order accu-rate finite difference schemes for the Helmholtz equationMoreover the plus signs in front of the leading order error
Journal of Applied Mathematics 3
Table 1 Analysis of the number of elements needed for a dispersionerror of minus 119896 = 10minus4
terms signify that the finite difference approximations willlag throughout the domain compared with the exact solutionmeaning that the discrete wave number is overestimated asis clear from Table 1 whereas in the case of finite elementsthe discrete wave number is underestimated [4 5]
For dispersion free propagation one requires that boththe exact and discrete waves propagate with the exact wavenumber that is = 119896 which is possible only when theproduct 119896ℎ rarr 0 for both second- and fourth-order schemesas is evident from expressions (7)Therefore one is interestedin discretization schemes (such as finite difference schemes)in which the discrete dispersion relation is independent ofboth the mesh size ℎ and the wave number 119896 With this inmind we modify the standard second-order central finitedifference scheme for the Helmholtz equation (3) such thatthe new scheme provides the exact solution at the nodes ofthe grid meaning that ℎ = 119896ℎ at all nodes of the grid Forthis we replace with 119896 in the discrete Bloch wave propertydefined in (5) and obtain
119906119895+119899
= 119890119894119896ℎ119899
119906119895
forall119899 isin Z (8)
Using the above property in (3) and (4) gives
119906119895minus1
+ ((119896ℎ)2minus 2) 119906
119895+ 119906119895+1
= (2 cos (119896ℎ) + (119896ℎ)2 minus 2) 119906119895
(9)
or
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (10)
119906119895minus2
minus 16119906119895minus1
+ (30 minus 12ℎ21198962) 119906119895minus 16119906
119895+1+ 119906119895+2
= (2 cos (2119896ℎ) minus 32 cos (119896ℎ) + 30 minus 12(119896ℎ)2) 119906119895
(11)
or
119906119895minus2
minus 16119906119895minus1
minus (2 cos (2119896ℎ) minus 32 cos (119896ℎ)) 119906119895
minus16119906119895+1
+ 119906119895+2
= 0(12)
The scheme obtained in (10) using the Bloch wave propertyis already presented in texts [8 10] with alternative formu-lations Interestingly the above schemes (10) and (12) leadback to (3) and (4) if we do Taylor series of the middle nodecoefficientsminus2 cos(119896ℎ) andminus2 cos(2119896ℎ)+32 cos(119896ℎ) and keeponly the terms up to second order Furthermore inserting anontrivial solution of the form 119906
119895= 119890119894119895ℎ into (10) and (12)
we obtain = 119896 which means propagation is dispersion free(or equivalently these modified schemes provide the exact
solution at the nodes of the grid) In order to construct exactcentral finite difference schemes of all orders we follow anexpression given in the book of Cohen [1] and present thefollowing generalized expression
11990610158401015840(119909) + 119896
2119906 (119909) =
1199012
sum119894=1
120582119894119906119895minus119894
+ 119906119895+119894
minus 2 cos 119894119896ℎ119906119895
(119894ℎ)2
= 0
(13)
where 119901 = 2119899 forall119899 isin N and 1205821199012119894=1
is given by
120582119894=
1199012
prod119897=1 119897 =119894
1198972
1198972 minus 1198942(14)
with the coefficient of the central node being given by
minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
(15)
It is also evident from Table 1 that higher-order accurateschemes such as (4) provide better accuracy with a smallernumber of elements for the Helmholtz equation Howeverhigher-order schemes require a greater number of stencilpoints and consequently (119886) the bandwidth of the resultingmatrix increases which is computationally more expensive toinvert and (119887) the number of fictitious nodes increases for thenodes near to the boundary and on the boundary itself Letus reconsider (10)
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (16)which is valid at all nodes 119895 = 0 1 2 119873 only when wehave Dirichlet boundary conditions applied at the end nodes119895 = 0 and 119895 = 119873 Otherwise when the above scheme isapplied at the end nodes it gives us the fictitious nodes givenby
119906minus1= minus2 cos (119896ℎ) 119906
1minus 1199062
119906119873+1
= minus2 cos (119896ℎ) 119906119873minus 119906119873minus1
(17)
Consequently in order to have a unique solution of theresulting system one requires additional information to killthese fictitious nodesThis ismore challenging for the fourth-order scheme which will produce fictitious nodes whenapplied at the boundary nodes 119895 = 0 and 119895 = 119873 as wellas for the nodes 119895 = 1 and 119895 = 119873 minus 1 adjacent to theboundary Therefore compact schemes are constructed withthe aim of retaining high accuracy with a smaller number ofstencil points [17] These discrepancies can be avoided if onecan make standard forward and backward finite differenceschemes exact which we do in the following section
22 Modified Forward and Backward Finite DifferenceSchemes We now consider the first-order forward and back-ward finite difference approximations of the second deriva-tive given by [16]
11990610158401015840
119895asymp119906119895minus 2119906119895+1
+ 119906119895+2
ℎ2
11990610158401015840
119895asymp119906119895minus 2119906119895minus1
+ 119906119895minus2
ℎ2
(18)
4 Journal of Applied Mathematics
Inserting the above approximations in (1) and performingsimplifications we get the following stencil forms for theforward and backward schemes respectively
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 (19)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895= 0 (20)
Using (8) in (19) and (20) we obtain modified forward andbackward finite difference schemes given by
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 (21)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895= 0 (22)
Similarly one can make forward and backward schemes ofany order exact In Table 3 we made only those forward andbackward schemes which keep only three terms in the finitedifference stencil exact Concluding one can use any of thepresented schemes to have nodally exact solutions for theHelmholtz equation when the problem is posed along withDirichlet boundary conditions In the following section wepresent modified schemes when the problem has radiationboundary conditions
We now make the first-order derivative involved in theradiation boundary condition given by
1199061015840
119895minus 119894119896119906119895= 0 (23)
exact The second-order central finite difference approxima-tion of 1199061015840(119909) is [16]
1199061015840
119895asymp119906119895+1
minus 119906119895minus1
2ℎ(24)
which on inserting into (23) gives the following stencil
119906119895minus1
minus 2119894119896ℎ119906119895minus 119906119895+1
= 0 (25)
Inserting a nontrivial solution of the form 119906119895= 119890119894119895ℎ results in
ℎ minus 119896ℎ = +(119896ℎ)3
6+ sdot sdot sdot (26)
which is not dispersion free so in order to make it dispersionfree we make use of the Bloch wave property (8) and end upwith the following form
119906119895minus1
+ 2119894 sin (119896ℎ) 119906119895minus 119906119895+1
= 0 (27)
which satisfies minus 119896 = 0 Hence the modified central finitedifference scheme (27) provides an exact solution at the last119873th node of the grid This scheme (27) was also obtained byWong and Li [10] but with a different formulation Scheme
(27) has one fictitous node for 119895 = 119873 which needs to beavoided So we are required to solve (27) with (10) to get
minus119906119895minus1
+ (119894 sin (119896ℎ) minus cos (119896ℎ)) 119906119895= 0 (28)
Instead of performing this extra step one can constructforward and backward schemes for radiation boundaryconditions and for that we consider first-order forward andbackward finite difference schemes with the first derivativegiven by
1199061015840
119895(119909) asymp
119906119895+1
minus 119906119895
ℎ
1199061015840
119895(119909) asymp
119906119895minus 119906119895minus1
ℎ
(29)
Now inserting (29) into (23) gives standard forward andbackward finite difference schemes
minus (1 + 119894119896ℎ) 119906119895+ 119906119895+1
= 0
minus119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895= 0
(30)
Similarly inserting (29) into (23) together with (8) we obtainmodified forward and backward finite difference schemes forradiation boundary conditions (23) given by
In Table 4 we have listed coefficients for standard and modi-fied schemes for radiation boundary conditions (23) for cen-tral forward and backward schemes
4 A Note on the Implementation ofthe Modified Schemes
An interesting feature of the modified schemes is theireasy implementation as (119886) one does not need to write abrand new code but instead one just needs to replace thecoefficient of the 119895th node in the standard schemes withthe modified coefficient (119887) the modified schemes keep thesame bandwidth structure as one has in the case of standardschemes and therefore adds no cost to the implementationbut one obtains highly accurate results
5 Numerical Examples forOne-Dimensional Problems
In order to illustrate the superiority of the modified schemeswe solve (1) on Ω = (0 1) sub R The numerical error is mea-sured using the discrete ℓ
infinnorm defined by ℓ
infin= max
119895|119906119895minus
119906(119909119895)| 119895 = 0 1 2 119873 with 119906(119909
119895) representing the
analytical solution and 119906119895the computed numerical solution
Moreover 119873 denotes the number of grid points in a uni-formly spaced grid
Journal of Applied Mathematics 5
51 Dirichlet Boundary Conditions Applied at Both Ends Firstof all we solve (1) on Ω = (0 1) sub R along with Dirichletboundary conditions given by
= 0 for 119895 = 119873 minus 1 (modified forward)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895
= 0 forall119895 = 2 119873 minus 1 (standard backward)
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (standard backward)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895
= 0 forall119895 = 2 119873 minus 1 (modified backward)
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (modified backward)
(33)
In Table 5 the dispersion error for a fixed mesh of sizeℎ = 10minus2 and a broad range of wave numbers from 119896 = 1 to119896 = 1014 is given It is evident that in the case of standardschemes (3) (19) and (20) the dispersion error dependsupon the nondimensional wave number 119896ℎ (Table 7) As for119896ℎ = 10
minus2 all schemes (3) (19) and (20) provide good resultswhereas they fail for 119896ℎ ≫ 1 However modified schemes(10) (21) and (22) provide highly accurate results even for119896ℎ = 1012 and are consistent with the dispersion relation minus 119896 = 0 which is independent of both the wave number
119896 and the mesh size ℎ Fu [18] made an effort and constructeda scheme in which the dispersion error in the leading ordererror term was only independent of the wave number
52 Dirichlet Boundary Condition at Left End and RadiationBoundary Condition at Right End Propagating Wave Thistime we solve (1) onΩ = (0 1) sub R along with Dirichlet andradiation boundary conditions applied at the left and rightends respectively
For this problem we use both standard and modifiedcentral finite difference schemes (3) and (10) for the approx-imation of the second derivative present in the Helmholtzequation and use all modified schemes central forward andbackward for the first derivative present in (23) presentedin Section 3 The full systems for the modified and standardfinite difference schemes are (Table 2)
+ (minus2 cos (119896ℎ) + 2119894 sin (119896ℎ)) 119906119895
= 0 for 119895 = 119873 (modified central)
2119906119895minus1
+ (minus2 + (119896ℎ)2+ 2119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard central)
119906119895minus1
+ (minus2 cos (119896ℎ) + 119890119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (modified forward)
119906119895minus1
+ (minus2 + (119896ℎ)2+ 1 + 119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard forward)
minus 119906119895minus1
+ 119890minus119894119896ℎ
119906119895
= 0 for 119895 = 119873 (modified backward)
minus 119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (standard backward)
(36)
Once again the superiority of the modified schemescompared with standard schemes is evident from Table 6even for the case of radiation boundary conditions
6 Journal of Applied Mathematics
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
by conventional (or standard) schemes is prohibitivelylarge
Standard finite difference schemes have extensively beenused [6ndash10] with the intention of having a precise answerto the major question ldquoHow can dispersion be eliminated orreducedwith less computational expenserdquo for the simulationof wave propagation However a serious impediment ofthe standard schemes developed so far is that they mayproduce optimal results for wave propagation problems (119886)for low wave numbers or (119887) with very high computationalcost but completely fail for high wave number propagationsHence the motivation is obvious for the development ofdiscretization schemes which are less expensive and highlyaccurate and efficient for the propagation of waves with veryhigh wave numbers
In this regard Nehrbass et al [8] in 1998 gave anew dimension to the second-order central finite difference(CFD) scheme by redefining the usual second-order CFDapproximation with a central node of 2 cos(119896ℎ)+ℎ21198962 insteadof 2Themajor limitation of this work was that (119886) it requiredprior information about the general solution of theHelmholtzequation and (119887) it did not provide exact results for problemswith nonreflecting boundary conditions However the fasterconvergence of this scheme was of no comparison to thestandard CFD scheme especially when the problem is solvedwith Dirichlet boundary conditions Recently in 2010 theidea of Nehrbass et al was further taken up by Wong andLi in [10] They proposed a different approach to solve theHelmholtz equation and their construction did not requirethe use of the general solution but instead they used theHelmholtz equation itself recursively In addition to thatthey also made the first derivative present in the radiationboundary condition exactThemost important feature of thisnew scheme was that it works for low as well as for very highwave numbers without the common weakness of fine meshsize even with radiation boundary conditions
The salient features of both schemes are (119886) easy (cheap)implementation which only requires the replacement of thecentral node (119887) the sparse tridiagonal structure of thematrixis preserved as one has in the case of the standard second-order CFD scheme (119888) both schemes provide nodally exactvalues on the interior grid points for the Helmholtz equationat any wave number
In this paper we present a modified approach to havenodally exact approximations of first- and second-orderderivatives for the Helmholtz equation on uniform grids byusing the Bloch wave property This will give us dispersionfree results in one dimension and optimal results in higherdimensions With the help of this alternate approach onecan make finite difference (central as well as forward andbackward) approximations of any order exact
The organization of the paper is as follows In Sections2 and 3 modified schemes are presented for the Helmholtzequation and radiation boundary conditions respectivelyIn Section 4 the easy (cheap) implementation of modifiedschemes is discussed In Section 6 modified schemes arepresented for two-dimensional problems Sections 5 and 6are devoted to numerical examples for both one- and two-dimensional problems respectively
In order to motivate the ideas we consider the one-dimen-sional Helmholtz equation [2]
11990610158401015840(119909) + 119896
2119906 (119909) = 0 for 119909 isin R (1)
where 119896 isin C is the wave number
21 Modified Central Finite Difference (CFD) Schemes Thestandard second- and fourth-order central finite differenceapproximations of the second-order derivative at the node119909119895= 119895ℎ of the uniformly spaced grid ℎZ are as follows [6 16]
11990610158401015840
119895asymp119906119895minus1
minus 2119906119895+ 119906119895+1
ℎ2
11990610158401015840
119895asympminus119906119895minus2
+ 16119906119895minus1
minus 30119906119895+ 16119906
119895+1minus 119906119895+2
12ℎ2
(2)
with ℎ gt 0 being the distance between two adjacent nodes ofthe grid ℎZWith these approximations for (1) we obtain thefollowing stencils
119906119895minus1
+ ((119896ℎ)2minus 2) 119906
119895+ 119906119895+1
= 0 (3)
119906119895minus2
minus 16119906119895minus1
+ (30 minus 12(119896ℎ)2) 119906119895minus 16119906
119895+1+ 119906119895+2
= 0 (4)
The above finite difference approximations (3) and (4) of theHelmholtz equation admit nontrivial solutions of the form119906119895= 119890119894119895ℎ which satisfy the property
119906119895+119899
= 119890119894ℎ119899
119906119895
forall119899 isin Z (5)
which is known as the discrete Bloch wave property [4]Moreover is the discrete wave number provided that itsatisfies
ℎ = cosminus1 (1 minus (119896ℎ)2
2)
ℎ = cosminus1 (4 minus radic9 minus 3(119896ℎ)2)
(6)
or
ℎ minus 119896ℎ = +(119896ℎ)3
24+ sdot sdot sdot
ℎ minus 119896ℎ = +(119896ℎ)5
180+ sdot sdot sdot
(7)
which we get upon writing the above expressions for ℎ as aseries in 119896ℎ
It is evident from (7) that second- and fourth-orderapproximations of the second derivative present in theHelmholtz equation provide second- and fourth-order accu-rate finite difference schemes for the Helmholtz equationMoreover the plus signs in front of the leading order error
Journal of Applied Mathematics 3
Table 1 Analysis of the number of elements needed for a dispersionerror of minus 119896 = 10minus4
terms signify that the finite difference approximations willlag throughout the domain compared with the exact solutionmeaning that the discrete wave number is overestimated asis clear from Table 1 whereas in the case of finite elementsthe discrete wave number is underestimated [4 5]
For dispersion free propagation one requires that boththe exact and discrete waves propagate with the exact wavenumber that is = 119896 which is possible only when theproduct 119896ℎ rarr 0 for both second- and fourth-order schemesas is evident from expressions (7)Therefore one is interestedin discretization schemes (such as finite difference schemes)in which the discrete dispersion relation is independent ofboth the mesh size ℎ and the wave number 119896 With this inmind we modify the standard second-order central finitedifference scheme for the Helmholtz equation (3) such thatthe new scheme provides the exact solution at the nodes ofthe grid meaning that ℎ = 119896ℎ at all nodes of the grid Forthis we replace with 119896 in the discrete Bloch wave propertydefined in (5) and obtain
119906119895+119899
= 119890119894119896ℎ119899
119906119895
forall119899 isin Z (8)
Using the above property in (3) and (4) gives
119906119895minus1
+ ((119896ℎ)2minus 2) 119906
119895+ 119906119895+1
= (2 cos (119896ℎ) + (119896ℎ)2 minus 2) 119906119895
(9)
or
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (10)
119906119895minus2
minus 16119906119895minus1
+ (30 minus 12ℎ21198962) 119906119895minus 16119906
119895+1+ 119906119895+2
= (2 cos (2119896ℎ) minus 32 cos (119896ℎ) + 30 minus 12(119896ℎ)2) 119906119895
(11)
or
119906119895minus2
minus 16119906119895minus1
minus (2 cos (2119896ℎ) minus 32 cos (119896ℎ)) 119906119895
minus16119906119895+1
+ 119906119895+2
= 0(12)
The scheme obtained in (10) using the Bloch wave propertyis already presented in texts [8 10] with alternative formu-lations Interestingly the above schemes (10) and (12) leadback to (3) and (4) if we do Taylor series of the middle nodecoefficientsminus2 cos(119896ℎ) andminus2 cos(2119896ℎ)+32 cos(119896ℎ) and keeponly the terms up to second order Furthermore inserting anontrivial solution of the form 119906
119895= 119890119894119895ℎ into (10) and (12)
we obtain = 119896 which means propagation is dispersion free(or equivalently these modified schemes provide the exact
solution at the nodes of the grid) In order to construct exactcentral finite difference schemes of all orders we follow anexpression given in the book of Cohen [1] and present thefollowing generalized expression
11990610158401015840(119909) + 119896
2119906 (119909) =
1199012
sum119894=1
120582119894119906119895minus119894
+ 119906119895+119894
minus 2 cos 119894119896ℎ119906119895
(119894ℎ)2
= 0
(13)
where 119901 = 2119899 forall119899 isin N and 1205821199012119894=1
is given by
120582119894=
1199012
prod119897=1 119897 =119894
1198972
1198972 minus 1198942(14)
with the coefficient of the central node being given by
minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
(15)
It is also evident from Table 1 that higher-order accurateschemes such as (4) provide better accuracy with a smallernumber of elements for the Helmholtz equation Howeverhigher-order schemes require a greater number of stencilpoints and consequently (119886) the bandwidth of the resultingmatrix increases which is computationally more expensive toinvert and (119887) the number of fictitious nodes increases for thenodes near to the boundary and on the boundary itself Letus reconsider (10)
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (16)which is valid at all nodes 119895 = 0 1 2 119873 only when wehave Dirichlet boundary conditions applied at the end nodes119895 = 0 and 119895 = 119873 Otherwise when the above scheme isapplied at the end nodes it gives us the fictitious nodes givenby
119906minus1= minus2 cos (119896ℎ) 119906
1minus 1199062
119906119873+1
= minus2 cos (119896ℎ) 119906119873minus 119906119873minus1
(17)
Consequently in order to have a unique solution of theresulting system one requires additional information to killthese fictitious nodesThis ismore challenging for the fourth-order scheme which will produce fictitious nodes whenapplied at the boundary nodes 119895 = 0 and 119895 = 119873 as wellas for the nodes 119895 = 1 and 119895 = 119873 minus 1 adjacent to theboundary Therefore compact schemes are constructed withthe aim of retaining high accuracy with a smaller number ofstencil points [17] These discrepancies can be avoided if onecan make standard forward and backward finite differenceschemes exact which we do in the following section
22 Modified Forward and Backward Finite DifferenceSchemes We now consider the first-order forward and back-ward finite difference approximations of the second deriva-tive given by [16]
11990610158401015840
119895asymp119906119895minus 2119906119895+1
+ 119906119895+2
ℎ2
11990610158401015840
119895asymp119906119895minus 2119906119895minus1
+ 119906119895minus2
ℎ2
(18)
4 Journal of Applied Mathematics
Inserting the above approximations in (1) and performingsimplifications we get the following stencil forms for theforward and backward schemes respectively
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 (19)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895= 0 (20)
Using (8) in (19) and (20) we obtain modified forward andbackward finite difference schemes given by
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 (21)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895= 0 (22)
Similarly one can make forward and backward schemes ofany order exact In Table 3 we made only those forward andbackward schemes which keep only three terms in the finitedifference stencil exact Concluding one can use any of thepresented schemes to have nodally exact solutions for theHelmholtz equation when the problem is posed along withDirichlet boundary conditions In the following section wepresent modified schemes when the problem has radiationboundary conditions
We now make the first-order derivative involved in theradiation boundary condition given by
1199061015840
119895minus 119894119896119906119895= 0 (23)
exact The second-order central finite difference approxima-tion of 1199061015840(119909) is [16]
1199061015840
119895asymp119906119895+1
minus 119906119895minus1
2ℎ(24)
which on inserting into (23) gives the following stencil
119906119895minus1
minus 2119894119896ℎ119906119895minus 119906119895+1
= 0 (25)
Inserting a nontrivial solution of the form 119906119895= 119890119894119895ℎ results in
ℎ minus 119896ℎ = +(119896ℎ)3
6+ sdot sdot sdot (26)
which is not dispersion free so in order to make it dispersionfree we make use of the Bloch wave property (8) and end upwith the following form
119906119895minus1
+ 2119894 sin (119896ℎ) 119906119895minus 119906119895+1
= 0 (27)
which satisfies minus 119896 = 0 Hence the modified central finitedifference scheme (27) provides an exact solution at the last119873th node of the grid This scheme (27) was also obtained byWong and Li [10] but with a different formulation Scheme
(27) has one fictitous node for 119895 = 119873 which needs to beavoided So we are required to solve (27) with (10) to get
minus119906119895minus1
+ (119894 sin (119896ℎ) minus cos (119896ℎ)) 119906119895= 0 (28)
Instead of performing this extra step one can constructforward and backward schemes for radiation boundaryconditions and for that we consider first-order forward andbackward finite difference schemes with the first derivativegiven by
1199061015840
119895(119909) asymp
119906119895+1
minus 119906119895
ℎ
1199061015840
119895(119909) asymp
119906119895minus 119906119895minus1
ℎ
(29)
Now inserting (29) into (23) gives standard forward andbackward finite difference schemes
minus (1 + 119894119896ℎ) 119906119895+ 119906119895+1
= 0
minus119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895= 0
(30)
Similarly inserting (29) into (23) together with (8) we obtainmodified forward and backward finite difference schemes forradiation boundary conditions (23) given by
In Table 4 we have listed coefficients for standard and modi-fied schemes for radiation boundary conditions (23) for cen-tral forward and backward schemes
4 A Note on the Implementation ofthe Modified Schemes
An interesting feature of the modified schemes is theireasy implementation as (119886) one does not need to write abrand new code but instead one just needs to replace thecoefficient of the 119895th node in the standard schemes withthe modified coefficient (119887) the modified schemes keep thesame bandwidth structure as one has in the case of standardschemes and therefore adds no cost to the implementationbut one obtains highly accurate results
5 Numerical Examples forOne-Dimensional Problems
In order to illustrate the superiority of the modified schemeswe solve (1) on Ω = (0 1) sub R The numerical error is mea-sured using the discrete ℓ
infinnorm defined by ℓ
infin= max
119895|119906119895minus
119906(119909119895)| 119895 = 0 1 2 119873 with 119906(119909
119895) representing the
analytical solution and 119906119895the computed numerical solution
Moreover 119873 denotes the number of grid points in a uni-formly spaced grid
Journal of Applied Mathematics 5
51 Dirichlet Boundary Conditions Applied at Both Ends Firstof all we solve (1) on Ω = (0 1) sub R along with Dirichletboundary conditions given by
= 0 for 119895 = 119873 minus 1 (modified forward)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895
= 0 forall119895 = 2 119873 minus 1 (standard backward)
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (standard backward)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895
= 0 forall119895 = 2 119873 minus 1 (modified backward)
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (modified backward)
(33)
In Table 5 the dispersion error for a fixed mesh of sizeℎ = 10minus2 and a broad range of wave numbers from 119896 = 1 to119896 = 1014 is given It is evident that in the case of standardschemes (3) (19) and (20) the dispersion error dependsupon the nondimensional wave number 119896ℎ (Table 7) As for119896ℎ = 10
minus2 all schemes (3) (19) and (20) provide good resultswhereas they fail for 119896ℎ ≫ 1 However modified schemes(10) (21) and (22) provide highly accurate results even for119896ℎ = 1012 and are consistent with the dispersion relation minus 119896 = 0 which is independent of both the wave number
119896 and the mesh size ℎ Fu [18] made an effort and constructeda scheme in which the dispersion error in the leading ordererror term was only independent of the wave number
52 Dirichlet Boundary Condition at Left End and RadiationBoundary Condition at Right End Propagating Wave Thistime we solve (1) onΩ = (0 1) sub R along with Dirichlet andradiation boundary conditions applied at the left and rightends respectively
For this problem we use both standard and modifiedcentral finite difference schemes (3) and (10) for the approx-imation of the second derivative present in the Helmholtzequation and use all modified schemes central forward andbackward for the first derivative present in (23) presentedin Section 3 The full systems for the modified and standardfinite difference schemes are (Table 2)
+ (minus2 cos (119896ℎ) + 2119894 sin (119896ℎ)) 119906119895
= 0 for 119895 = 119873 (modified central)
2119906119895minus1
+ (minus2 + (119896ℎ)2+ 2119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard central)
119906119895minus1
+ (minus2 cos (119896ℎ) + 119890119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (modified forward)
119906119895minus1
+ (minus2 + (119896ℎ)2+ 1 + 119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard forward)
minus 119906119895minus1
+ 119890minus119894119896ℎ
119906119895
= 0 for 119895 = 119873 (modified backward)
minus 119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (standard backward)
(36)
Once again the superiority of the modified schemescompared with standard schemes is evident from Table 6even for the case of radiation boundary conditions
6 Journal of Applied Mathematics
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
terms signify that the finite difference approximations willlag throughout the domain compared with the exact solutionmeaning that the discrete wave number is overestimated asis clear from Table 1 whereas in the case of finite elementsthe discrete wave number is underestimated [4 5]
For dispersion free propagation one requires that boththe exact and discrete waves propagate with the exact wavenumber that is = 119896 which is possible only when theproduct 119896ℎ rarr 0 for both second- and fourth-order schemesas is evident from expressions (7)Therefore one is interestedin discretization schemes (such as finite difference schemes)in which the discrete dispersion relation is independent ofboth the mesh size ℎ and the wave number 119896 With this inmind we modify the standard second-order central finitedifference scheme for the Helmholtz equation (3) such thatthe new scheme provides the exact solution at the nodes ofthe grid meaning that ℎ = 119896ℎ at all nodes of the grid Forthis we replace with 119896 in the discrete Bloch wave propertydefined in (5) and obtain
119906119895+119899
= 119890119894119896ℎ119899
119906119895
forall119899 isin Z (8)
Using the above property in (3) and (4) gives
119906119895minus1
+ ((119896ℎ)2minus 2) 119906
119895+ 119906119895+1
= (2 cos (119896ℎ) + (119896ℎ)2 minus 2) 119906119895
(9)
or
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (10)
119906119895minus2
minus 16119906119895minus1
+ (30 minus 12ℎ21198962) 119906119895minus 16119906
119895+1+ 119906119895+2
= (2 cos (2119896ℎ) minus 32 cos (119896ℎ) + 30 minus 12(119896ℎ)2) 119906119895
(11)
or
119906119895minus2
minus 16119906119895minus1
minus (2 cos (2119896ℎ) minus 32 cos (119896ℎ)) 119906119895
minus16119906119895+1
+ 119906119895+2
= 0(12)
The scheme obtained in (10) using the Bloch wave propertyis already presented in texts [8 10] with alternative formu-lations Interestingly the above schemes (10) and (12) leadback to (3) and (4) if we do Taylor series of the middle nodecoefficientsminus2 cos(119896ℎ) andminus2 cos(2119896ℎ)+32 cos(119896ℎ) and keeponly the terms up to second order Furthermore inserting anontrivial solution of the form 119906
119895= 119890119894119895ℎ into (10) and (12)
we obtain = 119896 which means propagation is dispersion free(or equivalently these modified schemes provide the exact
solution at the nodes of the grid) In order to construct exactcentral finite difference schemes of all orders we follow anexpression given in the book of Cohen [1] and present thefollowing generalized expression
11990610158401015840(119909) + 119896
2119906 (119909) =
1199012
sum119894=1
120582119894119906119895minus119894
+ 119906119895+119894
minus 2 cos 119894119896ℎ119906119895
(119894ℎ)2
= 0
(13)
where 119901 = 2119899 forall119899 isin N and 1205821199012119894=1
is given by
120582119894=
1199012
prod119897=1 119897 =119894
1198972
1198972 minus 1198942(14)
with the coefficient of the central node being given by
minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
(15)
It is also evident from Table 1 that higher-order accurateschemes such as (4) provide better accuracy with a smallernumber of elements for the Helmholtz equation Howeverhigher-order schemes require a greater number of stencilpoints and consequently (119886) the bandwidth of the resultingmatrix increases which is computationally more expensive toinvert and (119887) the number of fictitious nodes increases for thenodes near to the boundary and on the boundary itself Letus reconsider (10)
119906119895minus1
minus 2 cos (119896ℎ) 119906119895+ 119906119895+1
= 0 (16)which is valid at all nodes 119895 = 0 1 2 119873 only when wehave Dirichlet boundary conditions applied at the end nodes119895 = 0 and 119895 = 119873 Otherwise when the above scheme isapplied at the end nodes it gives us the fictitious nodes givenby
119906minus1= minus2 cos (119896ℎ) 119906
1minus 1199062
119906119873+1
= minus2 cos (119896ℎ) 119906119873minus 119906119873minus1
(17)
Consequently in order to have a unique solution of theresulting system one requires additional information to killthese fictitious nodesThis ismore challenging for the fourth-order scheme which will produce fictitious nodes whenapplied at the boundary nodes 119895 = 0 and 119895 = 119873 as wellas for the nodes 119895 = 1 and 119895 = 119873 minus 1 adjacent to theboundary Therefore compact schemes are constructed withthe aim of retaining high accuracy with a smaller number ofstencil points [17] These discrepancies can be avoided if onecan make standard forward and backward finite differenceschemes exact which we do in the following section
22 Modified Forward and Backward Finite DifferenceSchemes We now consider the first-order forward and back-ward finite difference approximations of the second deriva-tive given by [16]
11990610158401015840
119895asymp119906119895minus 2119906119895+1
+ 119906119895+2
ℎ2
11990610158401015840
119895asymp119906119895minus 2119906119895minus1
+ 119906119895minus2
ℎ2
(18)
4 Journal of Applied Mathematics
Inserting the above approximations in (1) and performingsimplifications we get the following stencil forms for theforward and backward schemes respectively
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 (19)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895= 0 (20)
Using (8) in (19) and (20) we obtain modified forward andbackward finite difference schemes given by
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 (21)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895= 0 (22)
Similarly one can make forward and backward schemes ofany order exact In Table 3 we made only those forward andbackward schemes which keep only three terms in the finitedifference stencil exact Concluding one can use any of thepresented schemes to have nodally exact solutions for theHelmholtz equation when the problem is posed along withDirichlet boundary conditions In the following section wepresent modified schemes when the problem has radiationboundary conditions
We now make the first-order derivative involved in theradiation boundary condition given by
1199061015840
119895minus 119894119896119906119895= 0 (23)
exact The second-order central finite difference approxima-tion of 1199061015840(119909) is [16]
1199061015840
119895asymp119906119895+1
minus 119906119895minus1
2ℎ(24)
which on inserting into (23) gives the following stencil
119906119895minus1
minus 2119894119896ℎ119906119895minus 119906119895+1
= 0 (25)
Inserting a nontrivial solution of the form 119906119895= 119890119894119895ℎ results in
ℎ minus 119896ℎ = +(119896ℎ)3
6+ sdot sdot sdot (26)
which is not dispersion free so in order to make it dispersionfree we make use of the Bloch wave property (8) and end upwith the following form
119906119895minus1
+ 2119894 sin (119896ℎ) 119906119895minus 119906119895+1
= 0 (27)
which satisfies minus 119896 = 0 Hence the modified central finitedifference scheme (27) provides an exact solution at the last119873th node of the grid This scheme (27) was also obtained byWong and Li [10] but with a different formulation Scheme
(27) has one fictitous node for 119895 = 119873 which needs to beavoided So we are required to solve (27) with (10) to get
minus119906119895minus1
+ (119894 sin (119896ℎ) minus cos (119896ℎ)) 119906119895= 0 (28)
Instead of performing this extra step one can constructforward and backward schemes for radiation boundaryconditions and for that we consider first-order forward andbackward finite difference schemes with the first derivativegiven by
1199061015840
119895(119909) asymp
119906119895+1
minus 119906119895
ℎ
1199061015840
119895(119909) asymp
119906119895minus 119906119895minus1
ℎ
(29)
Now inserting (29) into (23) gives standard forward andbackward finite difference schemes
minus (1 + 119894119896ℎ) 119906119895+ 119906119895+1
= 0
minus119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895= 0
(30)
Similarly inserting (29) into (23) together with (8) we obtainmodified forward and backward finite difference schemes forradiation boundary conditions (23) given by
In Table 4 we have listed coefficients for standard and modi-fied schemes for radiation boundary conditions (23) for cen-tral forward and backward schemes
4 A Note on the Implementation ofthe Modified Schemes
An interesting feature of the modified schemes is theireasy implementation as (119886) one does not need to write abrand new code but instead one just needs to replace thecoefficient of the 119895th node in the standard schemes withthe modified coefficient (119887) the modified schemes keep thesame bandwidth structure as one has in the case of standardschemes and therefore adds no cost to the implementationbut one obtains highly accurate results
5 Numerical Examples forOne-Dimensional Problems
In order to illustrate the superiority of the modified schemeswe solve (1) on Ω = (0 1) sub R The numerical error is mea-sured using the discrete ℓ
infinnorm defined by ℓ
infin= max
119895|119906119895minus
119906(119909119895)| 119895 = 0 1 2 119873 with 119906(119909
119895) representing the
analytical solution and 119906119895the computed numerical solution
Moreover 119873 denotes the number of grid points in a uni-formly spaced grid
Journal of Applied Mathematics 5
51 Dirichlet Boundary Conditions Applied at Both Ends Firstof all we solve (1) on Ω = (0 1) sub R along with Dirichletboundary conditions given by
= 0 for 119895 = 119873 minus 1 (modified forward)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895
= 0 forall119895 = 2 119873 minus 1 (standard backward)
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (standard backward)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895
= 0 forall119895 = 2 119873 minus 1 (modified backward)
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (modified backward)
(33)
In Table 5 the dispersion error for a fixed mesh of sizeℎ = 10minus2 and a broad range of wave numbers from 119896 = 1 to119896 = 1014 is given It is evident that in the case of standardschemes (3) (19) and (20) the dispersion error dependsupon the nondimensional wave number 119896ℎ (Table 7) As for119896ℎ = 10
minus2 all schemes (3) (19) and (20) provide good resultswhereas they fail for 119896ℎ ≫ 1 However modified schemes(10) (21) and (22) provide highly accurate results even for119896ℎ = 1012 and are consistent with the dispersion relation minus 119896 = 0 which is independent of both the wave number
119896 and the mesh size ℎ Fu [18] made an effort and constructeda scheme in which the dispersion error in the leading ordererror term was only independent of the wave number
52 Dirichlet Boundary Condition at Left End and RadiationBoundary Condition at Right End Propagating Wave Thistime we solve (1) onΩ = (0 1) sub R along with Dirichlet andradiation boundary conditions applied at the left and rightends respectively
For this problem we use both standard and modifiedcentral finite difference schemes (3) and (10) for the approx-imation of the second derivative present in the Helmholtzequation and use all modified schemes central forward andbackward for the first derivative present in (23) presentedin Section 3 The full systems for the modified and standardfinite difference schemes are (Table 2)
+ (minus2 cos (119896ℎ) + 2119894 sin (119896ℎ)) 119906119895
= 0 for 119895 = 119873 (modified central)
2119906119895minus1
+ (minus2 + (119896ℎ)2+ 2119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard central)
119906119895minus1
+ (minus2 cos (119896ℎ) + 119890119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (modified forward)
119906119895minus1
+ (minus2 + (119896ℎ)2+ 1 + 119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard forward)
minus 119906119895minus1
+ 119890minus119894119896ℎ
119906119895
= 0 for 119895 = 119873 (modified backward)
minus 119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (standard backward)
(36)
Once again the superiority of the modified schemescompared with standard schemes is evident from Table 6even for the case of radiation boundary conditions
6 Journal of Applied Mathematics
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
Inserting the above approximations in (1) and performingsimplifications we get the following stencil forms for theforward and backward schemes respectively
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 (19)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895= 0 (20)
Using (8) in (19) and (20) we obtain modified forward andbackward finite difference schemes given by
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 (21)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895= 0 (22)
Similarly one can make forward and backward schemes ofany order exact In Table 3 we made only those forward andbackward schemes which keep only three terms in the finitedifference stencil exact Concluding one can use any of thepresented schemes to have nodally exact solutions for theHelmholtz equation when the problem is posed along withDirichlet boundary conditions In the following section wepresent modified schemes when the problem has radiationboundary conditions
We now make the first-order derivative involved in theradiation boundary condition given by
1199061015840
119895minus 119894119896119906119895= 0 (23)
exact The second-order central finite difference approxima-tion of 1199061015840(119909) is [16]
1199061015840
119895asymp119906119895+1
minus 119906119895minus1
2ℎ(24)
which on inserting into (23) gives the following stencil
119906119895minus1
minus 2119894119896ℎ119906119895minus 119906119895+1
= 0 (25)
Inserting a nontrivial solution of the form 119906119895= 119890119894119895ℎ results in
ℎ minus 119896ℎ = +(119896ℎ)3
6+ sdot sdot sdot (26)
which is not dispersion free so in order to make it dispersionfree we make use of the Bloch wave property (8) and end upwith the following form
119906119895minus1
+ 2119894 sin (119896ℎ) 119906119895minus 119906119895+1
= 0 (27)
which satisfies minus 119896 = 0 Hence the modified central finitedifference scheme (27) provides an exact solution at the last119873th node of the grid This scheme (27) was also obtained byWong and Li [10] but with a different formulation Scheme
(27) has one fictitous node for 119895 = 119873 which needs to beavoided So we are required to solve (27) with (10) to get
minus119906119895minus1
+ (119894 sin (119896ℎ) minus cos (119896ℎ)) 119906119895= 0 (28)
Instead of performing this extra step one can constructforward and backward schemes for radiation boundaryconditions and for that we consider first-order forward andbackward finite difference schemes with the first derivativegiven by
1199061015840
119895(119909) asymp
119906119895+1
minus 119906119895
ℎ
1199061015840
119895(119909) asymp
119906119895minus 119906119895minus1
ℎ
(29)
Now inserting (29) into (23) gives standard forward andbackward finite difference schemes
minus (1 + 119894119896ℎ) 119906119895+ 119906119895+1
= 0
minus119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895= 0
(30)
Similarly inserting (29) into (23) together with (8) we obtainmodified forward and backward finite difference schemes forradiation boundary conditions (23) given by
In Table 4 we have listed coefficients for standard and modi-fied schemes for radiation boundary conditions (23) for cen-tral forward and backward schemes
4 A Note on the Implementation ofthe Modified Schemes
An interesting feature of the modified schemes is theireasy implementation as (119886) one does not need to write abrand new code but instead one just needs to replace thecoefficient of the 119895th node in the standard schemes withthe modified coefficient (119887) the modified schemes keep thesame bandwidth structure as one has in the case of standardschemes and therefore adds no cost to the implementationbut one obtains highly accurate results
5 Numerical Examples forOne-Dimensional Problems
In order to illustrate the superiority of the modified schemeswe solve (1) on Ω = (0 1) sub R The numerical error is mea-sured using the discrete ℓ
infinnorm defined by ℓ
infin= max
119895|119906119895minus
119906(119909119895)| 119895 = 0 1 2 119873 with 119906(119909
119895) representing the
analytical solution and 119906119895the computed numerical solution
Moreover 119873 denotes the number of grid points in a uni-formly spaced grid
Journal of Applied Mathematics 5
51 Dirichlet Boundary Conditions Applied at Both Ends Firstof all we solve (1) on Ω = (0 1) sub R along with Dirichletboundary conditions given by
= 0 for 119895 = 119873 minus 1 (modified forward)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895
= 0 forall119895 = 2 119873 minus 1 (standard backward)
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (standard backward)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895
= 0 forall119895 = 2 119873 minus 1 (modified backward)
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (modified backward)
(33)
In Table 5 the dispersion error for a fixed mesh of sizeℎ = 10minus2 and a broad range of wave numbers from 119896 = 1 to119896 = 1014 is given It is evident that in the case of standardschemes (3) (19) and (20) the dispersion error dependsupon the nondimensional wave number 119896ℎ (Table 7) As for119896ℎ = 10
minus2 all schemes (3) (19) and (20) provide good resultswhereas they fail for 119896ℎ ≫ 1 However modified schemes(10) (21) and (22) provide highly accurate results even for119896ℎ = 1012 and are consistent with the dispersion relation minus 119896 = 0 which is independent of both the wave number
119896 and the mesh size ℎ Fu [18] made an effort and constructeda scheme in which the dispersion error in the leading ordererror term was only independent of the wave number
52 Dirichlet Boundary Condition at Left End and RadiationBoundary Condition at Right End Propagating Wave Thistime we solve (1) onΩ = (0 1) sub R along with Dirichlet andradiation boundary conditions applied at the left and rightends respectively
For this problem we use both standard and modifiedcentral finite difference schemes (3) and (10) for the approx-imation of the second derivative present in the Helmholtzequation and use all modified schemes central forward andbackward for the first derivative present in (23) presentedin Section 3 The full systems for the modified and standardfinite difference schemes are (Table 2)
+ (minus2 cos (119896ℎ) + 2119894 sin (119896ℎ)) 119906119895
= 0 for 119895 = 119873 (modified central)
2119906119895minus1
+ (minus2 + (119896ℎ)2+ 2119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard central)
119906119895minus1
+ (minus2 cos (119896ℎ) + 119890119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (modified forward)
119906119895minus1
+ (minus2 + (119896ℎ)2+ 1 + 119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard forward)
minus 119906119895minus1
+ 119890minus119894119896ℎ
119906119895
= 0 for 119895 = 119873 (modified backward)
minus 119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (standard backward)
(36)
Once again the superiority of the modified schemescompared with standard schemes is evident from Table 6even for the case of radiation boundary conditions
6 Journal of Applied Mathematics
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
= 0 for 119895 = 119873 minus 1 (modified forward)
119906119895minus2
minus 2119906119895minus1
+ ((119896ℎ)2+ 1) 119906
119895
= 0 forall119895 = 2 119873 minus 1 (standard backward)
((119896ℎ)2+ 1) 119906
119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (standard backward)
119906119895minus2
minus 2119906119895minus1
+ (2119890minus119894119896ℎ
minus 119890minus2119894119896ℎ
) 119906119895
= 0 forall119895 = 2 119873 minus 1 (modified backward)
(2119890119894119896ℎ
minus 1198902119894119896ℎ
) 119906119895minus 2119906119895+1
+ 119906119895+2
= 0 for 119895 = 1 (modified backward)
(33)
In Table 5 the dispersion error for a fixed mesh of sizeℎ = 10minus2 and a broad range of wave numbers from 119896 = 1 to119896 = 1014 is given It is evident that in the case of standardschemes (3) (19) and (20) the dispersion error dependsupon the nondimensional wave number 119896ℎ (Table 7) As for119896ℎ = 10
minus2 all schemes (3) (19) and (20) provide good resultswhereas they fail for 119896ℎ ≫ 1 However modified schemes(10) (21) and (22) provide highly accurate results even for119896ℎ = 1012 and are consistent with the dispersion relation minus 119896 = 0 which is independent of both the wave number
119896 and the mesh size ℎ Fu [18] made an effort and constructeda scheme in which the dispersion error in the leading ordererror term was only independent of the wave number
52 Dirichlet Boundary Condition at Left End and RadiationBoundary Condition at Right End Propagating Wave Thistime we solve (1) onΩ = (0 1) sub R along with Dirichlet andradiation boundary conditions applied at the left and rightends respectively
For this problem we use both standard and modifiedcentral finite difference schemes (3) and (10) for the approx-imation of the second derivative present in the Helmholtzequation and use all modified schemes central forward andbackward for the first derivative present in (23) presentedin Section 3 The full systems for the modified and standardfinite difference schemes are (Table 2)
+ (minus2 cos (119896ℎ) + 2119894 sin (119896ℎ)) 119906119895
= 0 for 119895 = 119873 (modified central)
2119906119895minus1
+ (minus2 + (119896ℎ)2+ 2119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard central)
119906119895minus1
+ (minus2 cos (119896ℎ) + 119890119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (modified forward)
119906119895minus1
+ (minus2 + (119896ℎ)2+ 1 + 119894119896ℎ) 119906
119895
= 0 for 119895 = 119873 (standard forward)
minus 119906119895minus1
+ 119890minus119894119896ℎ
119906119895
= 0 for 119895 = 119873 (modified backward)
minus 119906119895minus1
+ (1 minus 119894119896ℎ) 119906119895
= 0 for 119895 = 119873 (standard backward)
(36)
Once again the superiority of the modified schemescompared with standard schemes is evident from Table 6even for the case of radiation boundary conditions
6 Journal of Applied Mathematics
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
Table 2The coefficients of the central nodes for modified and standard schemes where the coefficient for standard schemes can be obtainedfrom the series representations of the central node of the modified scheme
Order Modified CFD scheme Standard CFD scheme
2 minus2 cos (119896ℎ)ℎ2
minus2 + (119896ℎ)2
ℎ2
4 2 cos (2119896ℎ) minus 32 cos (119896ℎ)12ℎ2
30 minus 12(119896ℎ)2
12ℎ2
6 minus4 cos (3119896ℎ) + 5 cos (2119896ℎ) minus 540 cos (119896ℎ)180ℎ2
minus490 + 180(119896ℎ)2
180ℎ2
2119901 minus2
1199012
sum119894=1
120582119894
cos (119894119896ℎ)(119894ℎ)2
with 120582119894=
1199012
prod119897=1119897 =119894
1198972
1198972 minus 1198942minus2
ℎ2
1199012
sum119894=1
120582119894
1198942(1 minus
(119894119896ℎ)2
2)
Table 3 Coefficients of the 119895th node for modified and standardforward and backward schemes for (1) where the coefficients forstandard schemes can be recovered from the real part of the seriesrepresentations of the modified coefficients
Modified coefficient Standard coefficient2119890119894119896ℎminus 1198902119894119896ℎ
1 + (119896ℎ)2
5119890119894119896ℎ minus 41198902119894119896ℎ + 1198903119894119896ℎ 2 + (119896ℎ)2
104119890119894119896ℎ minus 1141198902119894119896ℎ + 561198903119894119896ℎ minus 111198904119894119896ℎ 35 + 12(119896ℎ)2
Modified coefficient Standard coefficient2119890minus119894119896ℎ minus 119890minus2119894119896ℎ 1 + (119896ℎ)
2
5119890minus119894119896ℎ
minus 4119890minus2119894119896ℎ
+ 119890minus3119894119896ℎ
2 + (119896ℎ)2
104119890minus119894119896ℎ minus 114119890minus2119894119896ℎ + 56119890minus3119894119896ℎ minus 11119890minus4119894119896ℎ 35 + 12(119896ℎ)2
Table 4 Coefficients of the 119895th node for modified and standardcentral and forward and backward schemes for (23) where thecoefficients for standard schemes can be recovered from the real partof the series representations of the modified coefficients
Modified coefficient Standard coefficientCentral minus2 sin 119896ℎ minus2119894119896ℎ
Scheme minus119890minus2119894119896ℎ2 + 8119890minus119894119896ℎ minus 8119890119894119896ℎ + 1198902119894119896ℎ minus12119894119896ℎ
Forward minus119890119894119896ℎ
minus1 minus 119894119896ℎ
Scheme minus4119890119894119896ℎ + 1198902119894119896ℎ minus3 minus 2119894119896ℎ
Backward 119890minus119894119896ℎ 1 minus 119894119896ℎ
Scheme minus4119890minus119894119896ℎ + 119890minus2119894119896ℎ minus3 minus 2119894119896ℎ
6 Modified Schemes for the Two-DimensionalHelmholtz Equation on Square Meshes
We consider the two-dimensional Helmholtz equation givenby
119906119909119909+ 119906119910119910+ 1198962119906 (119909 119910) = 0 on R
2 (37)
with 1198962 = 1198962119909+ 1198962119910 where 119896
119909= 119896 cos 120579 and 119896
119910= 119896 sin 120579
are user-specified constants and 120579 is the incident angle Nowusing the following approximations
119906119909119909asymp119906119894minus1119895
minus 2119906119894119895+ 119906119894+1119895
ℎ2
119906119910119910asymp119906119894119895minus1
minus 2119906119894119895+ 119906119894119895+1
ℎ2
(38)
(37) takes the form
119906119894minus1119895
+ 119906119894+1119895
+ 119906119894119895minus1
+ 119906119894119895+1
+ (ℎ21198962minus 4) 119906
119894119895= 0 (39)
Equation (39) is the standard five-point second-order centralfinite difference scheme for (37) In order to construct anexact scheme for (37) we now use the following Bloch waveproperty for the two-dimensional case given by
which is the required exact central finite difference schemefor (37) Now in order to avoid duplication one can constructexact forward and backward finite difference schemes for (37)as explained in the case of the one-dimensional Helmholtzequation in Section 22
7 Numerical Examples forTwo-Dimensional Problems
We reconsider (37) defined on the square domain (0 1)2
In order to illustrate the effectiveness of the modified schemein comparison to the standard finite difference scheme wesolve (43) on a square mesh of mesh length ℎ gt 0 for two
Journal of Applied Mathematics 7
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
Table 5 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10minus2 with varying wave numbers for the case ofDirichlet boundary conditions (32)
Table 6 Comparison of the dispersion error for standard central (3) and modified central (10) standard forward (19) and modified forward(21) and backward (20) and modified backward (22) finite difference schemes for fixed ℎ = 10
minus2 with varying wave numbers for the case ofradiation boundary conditions (34)
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
Table 7 Comparison of the dispersion error at an angle of 1205874 for standard and modified finite difference schemes for fixed ℎ = 10minus2 withvarying wave numbers for the case of (45) and (46) boundary conditions
Dirichlet BCs (45) Dirichlet and radiation BCs (46)119896ℎ Standard Modified Standard Modified10minus6
The authors declare that there is no conflict of interestsregarding the publishing of this paper
References
[1] G C Cohen Higher-Order Numerical Methods for TransientWave Equations Scientific Computation Springer Berlin Ger-many 2002
[2] F Ihlenburg Finite Element Analysis of Acoustic Scattering vol132 of Applied Mathematical Sciences Springer New York NYUSA 1998
[3] R J LeVeque Finite Volume Methods for Hyperbolic ProblemsCambridge Texts in Applied Mathematics Cambridge Univer-sity Press Cambridge UK 2002
[4] M Ainsworth and H A Wajid ldquoExplicit discrete dispersionrelations for the acoustic wave equation in d-dimensionsusing finite element spectral element and optimally blendedschemesrdquo inComputerMethods inMechanics vol 1 ofAdvancedStructured Materials pp 3ndash17 Springer Berlin Germany 2004
[5] M Ainsworth and H A Wajid ldquoOptimally blended spectral-finite element scheme for wave propagation and nonstandardreduced integrationrdquo SIAM Journal on Numerical Analysis vol48 no 1 pp 346ndash371 2010
[6] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992
[7] M Nabavi M H K Siddiqui and J Dargahi ldquoA new 9-pointsixth-order accurate compact finite-difference method for theHelmholtz equationrdquo Journal of Sound and Vibration vol 307no 3ndash5 pp 972ndash982 2007
[8] JWNehrbass J O Jevtic andR Lee ldquoReducing the phase errorfor finite-difference methods without increasing the orderrdquoIEEE Transaction on Antennas and Propagation vol 46 no 8pp 1194ndash1201 1998
[9] I Singer and E Turkel ldquoHigh-order finite difference methodsfor the Helmholtz equationrdquo Computer Methods in AppliedMechanics and Engineering vol 163 no 1ndash4 pp 343ndash358 1998
[10] Y SWong andG Li ldquoExact finite difference schemes for solvingHelmholtz equation at any wavenumberrdquo International Journalof Numerical Analysis and Modeling B vol 2 no 1 pp 91ndash1082011
[11] M Ainsworth and H A Wajid ldquoDispersive and dissipativebehavior of the spectral element methodrdquo SIAM Journal onNumerical Analysis vol 47 no 5 pp 3910ndash3937 2009
[12] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagationmdashII Three-dimensional mod-els oceans rotation and self-gravitationrdquo Geophysical JournalInternational vol 150 no 1 pp 303ndash318 2002
[13] D Komatitsch and J Tromp ldquoSpectral-element simulations ofglobal seismic wave propagation - I Validationrdquo GeophysicalJournal International vol 149 no 2 pp 390ndash412 2002
[14] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number Part I The h-version of the FEMrdquo Computers amp Mathematics with Applica-tions vol 30 no 9 pp 9ndash37 1995
[15] F Ihlenburg and I Babuska ldquoFinite element solution of theHelmholtz equation with high wave number IIThe h-p version
Journal of Applied Mathematics 9
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
of the FEMrdquo SIAM Journal on Numerical Analysis vol 34 no 1pp 315ndash358 1997
[16] E H Twizell Computational Methods for Partial DifferentialEquations Ellis Horwood Series Mathematics and its Applica-tions Ellis Horwood Chichester UK 1984
[17] T W H Sheu L W Hsieh and C F Chen ldquoDevelopment ofa three-point sixth-order Helmholtz schemerdquo Journal of Com-putational Acoustics vol 16 no 3 pp 343ndash359 2008
[18] Y Fu ldquoCompact fourth-order finite difference schemes forHelmholtz equation with high wave numbersrdquo Journal ofComputational Mathematics vol 26 no 1 pp 98ndash111 2008
