joseph p. gerrity, jr. development division this is an ...€¦ · c set giuf.ss.to mean value of...
TRANSCRIPT
U.S. DEPARTMENT OF COMMERCENATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
NATIONAL WEATHER SERVICENATIONAL METEOROLOGICAL CENTER
OFFICE NOTE 180
A Method for Solving the Helmholtz Equation in aRectangular Region Using Homogeneous Neuman Boundary Conditions
Joseph P. Gerrity, Jr.Development Division
JUNE 1978
This is an unreviewed manuscript, primarilyintended for informal exchange of informationamong NMC staff members.
1. Introduction
The Helmholtz equation is
V24 - X(x,y)4 = G(x,y) (1)
where X is a real, non-negative function, V2 is the Laplacian, ~ is the
unknown, and G is a forcing function. The boundary conditions to be
satisfied are -bn= 0, where - is the outward normal derivative.
The numerical algorithm presented here for the solution of this
equation is a split-pseudo-direct method. We approximate the equation
(1) using finite differences. A recurrence, or relaxation, equation is
written to develop a sequence of solutions. Unlike the S0R method, we
use implicit approximations in the recurrence equation in one coordinate
direction. The implicit equation is inverted using Gaussian elimination.
After completing the generation of the new approximate solution the
process is repeated, but the other coordinate direction is now approxi-
mated implicitly in the recurrence equation.
Just two such passes have been found necessary for a test problem.
2. The finite difference equation
Assume that the region is rectangular of size ((P+l)Ax,(Q+l)Ax)-.
Equation (1) is written,
Oi+l,j + Oi-l,j + oi,j+l + i,j- (4 +ij) ij = Gij (2)
whereij = (Ax)2 X((i-l)Ax,(j-1)Ax)
andGij = (Ax)2 G((i-l)Ax,(j-l)Ax)
-2-
The recurrence (or relaxation) equation is written in step 1.
n+l n rn+l n+l n n+l n+l~ ~4i+~,.j + n + n+l~ - (4+Xij)4.' - G.)(3)~ij = ~ij [i+l,j + ~-l,j+ +~ ij + T ij-1 _j lj
In this step the i coordinate is done implicitly, The boundary con-
ditions are evoked wherever they can be used.
The equation (3) is rewritten
n~~l 1),~n+l n+l- an + (l+a(4+Xij)),j+l - a 4i+l = R.. (4)
where R.. n + a, (O n + on~~~~~~~~~~l (5)ij 1 J +l ijl ij- 1 ij)
The solution of (4) by Gaussian Elimination (cf. Richtmyer and
Morton, p. 198) is carried out, augmenting j after each solution. On the
row j=2, the variable ,jn+l is equal to n+l because of the boundary con-
dition, and this fact is used to modify (4). Similarly when j=Q, the
parameter ij+l is replaced by ni+l because of the boundary condition.
In step 2, the recurrence equation is modified to
n+lj = n + a (in+l + 4ij l + i n+l - (4+Xi) n+l -_ G.) (6)= 4. + i+l,j i-lj J ij j
The j-coordinate direction is now done implicitly. The modifications of
the system are similar to those outlined in step 1.
3. A test problem
In Appendix A, the Fortran code for a test problem is given. A 'true'
solution was generated using an IBM random number generator. The Helmholtz
term was similarly generated. Using these fields, the forcing function was
-3-
calculated for P=52, Q=46. Employing a first guess solution that was
everywhere equal to the mean value of the 'true' solution, we ran the
code with a=k. At no point did the numerical solution differ from the
'true' solution by 1 part in 1000.
The average value of the 'true' solution is 5496.098, that of the
numerical solution is 5496.094. The Bias was -.321E-2, and the Root-
mean-square error of the numerical solution was 0.365E-2.
4. Further work
The method will be explored further using larger domains and different
values of X, including zero.
Reference
Richtmyer and Morton, Difference methods for initial value problems.
Interscience, New York, N.Y., 1967.
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C. A tMl PROGRAM FOR TtIF SOLUTIOM OF THE HELMHOLTZ EQUATIONC
DItiENS[Tn Z7(53947)]TSOL(53,47),R(5347),HLN(53147)CC Z IS'THF 'I.PMERICAL RnLUITIOnM OF THF DTFFFRF.ICE Ent.IATIOTNC :)TSOL TS THF TP'IF SlL.jTI[ON OF THE DIFFERENCE EIUATIOr, IC . IS THF FnPCIIjG F!.F:tcTTON FOR TS13L,C HLM IS :THE HELrHOLTZ COEFFICIFNT.CC ::AL-'HA IS TIlE COEFFICIENT OF RELAXATION.
DATA AL.PHA/.25/C
*CALL I!!ITL(PTSOL·HLM)CC SET GIUF.SS.TO MEAN VALUE OF TSOL.C
FACT = 51*4- 5 .TrEAr = O.0 D 1 T=2,520- 1 J=2.46TrEAI = TMIEAN + TSOL(IJ)/FACT .
I COTI'JTTNUE- ::DC 2 1=1,53- .c J=11.47ZtT,.t) = TMEAN
2 CO']ITI1:7
PRTNT 3, Tr'EAN3 FORr"AT(1HO0,'AVERAGE VALUE OF TSOL=',F1lO3)
CC STEpI SOLVES THE ITMpLICTT ITEpATIVE En. IN THE X DITRECTIC
00ono CCrITT.IIjECALL STFI (7,R.HlLMALPHA',IER): .CALL rTFOI.!T4TSOL,Z,1 ) CALL STFP2( ,RHLMALPHA·IER)CALL DTFOUT(TSOLZ,2)IFTFER Eo O0) GO0 4PR I TNT 9
5 FORMAT(ItO0,HAVE TO STOP PRORLF'r IN STEP1')SToP
4 CONTNIUEC
C CALCIILATE STATISTICS OF SOLUTION
CALL STAT(Z7TSOL) STOP . .Ern N..- '
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11 CONiTINUIJE . :C SOL>'E EO*
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12 £.OhlT!~HUJE . -Z(1,2) ='Z('2*2? ' · . :Z(53,2) = 7(52,2) -DO 13 T:1,937(Tl) :Zf 7I 2)
13 CCNTIPJINUE . :C J =2 <( .... FND
C LOr,P TO Dn J=3 THIROuGH 45 ...-STAPTC
DO D0 J=3,-45.C : SFT -UP EQUATIOF! PARAMETERS
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33 CONPTIMIIF so' , Vr Eo
' -CALl.- [FELI.F(C,,0EF, Ar!S, TER 951) -IFi{ER.rcE.n) GO TO -o-00[0 31 T=2,52: 7(,J) = AFISII-1I.) -. . .
31. CONTTNUE .. .Z(1'J) = ?2,J)Z(53,J) = 7(52,J)
C 20 CONTINUEC
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40 COr4JTTN!i :DJO 41 I=1,51FtT) = 7(T+1,4(,) + · ALPHA ' I Z(I+,45) -- R(T+l946) )
41 COb!TIm'll: ;[C SnLVr FO.
CALL GFl. WECnEF, Ars, TER,51) -TIF(IFR.Frl.n) GO TO s9000 DO 42 I=2,:2Z(T476) = Z'(I-46) : : .
42 CONTTI!NJ UE: ;0 : (1:.14A) := Z(P*46): ;0. :: : : ; V
Z(53,4A) = 7(52,46);:.E S ; - iZII ,47 )- = ' ( 2,qr 1 2 0 -: :
' 7(- ,l47) = Z(52,46)
C COMPLETES SOLUTOPI ... ALL pTSC
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SLJ3ROUTINE STEP2(7,PHLMALPHAf,]RIR)'DI.PIEM'I-T N 7 (5?, 4 7),fR(53,47) ,L"(93,U7)DOURLE PRECISTON C(9:1),D1I,/r51),F(~1IhANS(51)r
C SET Up REL.XATTON, E.-C 1=2 .,. START
P0 1.0 .d=2,q6 '~F(J-1) = Z(2,dJ .. + ALPHA 7 Z(. J) - R(2,J)
lo CO.ITIrNUIJDO 11 J=1,45p(J)-= 1; + ALPHA * (.3. + HLM(2,,J+1)C(J) =- ALPHA.;' E (J) - ALPHAIF( J .Eq. 1 (OP. j *EO. 45! D(J) = D(J) + C(J)
11.. COTTJ.UEC SOLVE EQoCALL GCLITM(CfL[,EFA ,S',TER,45).
IFirFUP .Hr!r. ) GO: TO 9000anO 19 J='2t'f,7(2,J# = ANq(J-1)
12 CONTTNUE7¢2,1) = Z(2,2)Z·2*7) = *6'I3 J=i, ...
7(1,.J) = ZI2. J)13 CONTINIIE
C I =2 X.. ENDC
C. LOOp TO+ DO 1=3 THRU 51 ... START'C
[30 20-T=3,51C:' SFT UP EQUATION PARAM£ETERS
PC 30 J=],45.C(J) - ALPHAEl (J) = 1. + ALPHA * ( 4.. + HLM¢],+IE(j) = - ALPHAIF( J *tI. 1 'OR. J Eon. 45 1 0(J) = 1n(j) + C(J)FIJI 7=T?,+l) + ALPHA * (.Z(T+1e.J+) + Z(I-2,J+l) -R(TJ+i)
30 COITINHIESOL'E FO.CALl- GL'LFM(CD,FF,FA'S, TEP,45)Tr(IpR f'tFr. 0) GO TO 9000[)0 'zl J=2,46·'Z([,UJ : AI'SIJ-1)
31 CO.ITTNJUF?lI19I) = Z(IT2)Z(T.q47).= Z(I,4-6)
C20 CONTINUE
C
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421 CONTTNUEC-C SOLVE ECO. -
C:,"PcLL C.LIr(C,fEF,AmS,TER, 45 )TIF(IEP .!"E. 0 ) GO TO qOO 0;30 42 J=2,46
Z(52,J1 = AN$S J-l l I - -Z(53,J) = Z(52,J)
42 CONT IJCE:Z(5?,tI) = Z(52 ,2)7 ( 5?, 47)= 7(52 46) - .
.(f531) = Z(,2 " : 2 . .7Z(5947) = 7(52,46)
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T * t/VIS) (F), tERE T IS A TRI-DTAGOPlAL MATRIX OF ORDER Ng
An AE 'AA'!D F AE !", DT. ENSTIOMAL vr'CTOr'Sb CDA'TI;TIDE ARE THF tELEIL'TS OF THE [E.ATRIX T, NOTE Ctl&E(N) DO
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INTTIALTZE
: D(l)TF ( r .EO. 0,. GO TO 9000 .
SW = F{) / G
EL.TIT?,:ATION (FORWARD pROCESS)
G.=(V)'- l(K-1) *oC(K)IF( .frn. 0.) Go0 T_ 9000 ·It{K) = F(K) / cS(K) = I F(K) C(K) * S(K-1) ) / G
o CONITTNtIIE
LAST O!'FRS .. NOTE HN} = o0
= r ('i) -.(l) * H -:'-I)IF( r- .F. 0f.) GO TO 9n0-S{ql = ( F~t6) -CmiN * S(N-!} ) / r'
ncA.CK SluFSTITUTIONl
ro 'O K=1 ,IrdI' n n -T .k'
7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I'.tIT1 =-MI) t It (T) * AS {I+1) -o or4TTI.l IE
iER = 0RFTIURkN
)0*I[ ~ = 1RIETURNENMD
o::T I orJs TN .C NrFFEC *NArF( ',.",) NlooPTTMIZE LTNr..OUI!T ( 60 ) SIE (MAX) AIITODBL !ONF )
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LEVEL 2.2 , (SEPT 76)D OS/36 0 FORTRAN H FrYTENDEn PLusRE2UESTEn OPTIONS. 'NODEcKOr.O-ISTOPTTMIMrnn ,EMApSTZE (mAX) MNILFoyRFF ,MnTrRM'LCOPTION,]S rN EFFECT. NAr;F MAAT ,iN fionPTr'iTF TIrFCOUrIT(6n) ST7lr(MAXJ AITOJrLFnN :
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. . *O=TrTr-NS INr. 4lOPTIONS Ir]
._ *O.PTIONS IN
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*TASr.TISTICS*
0 ***** END C
:SUWPOUTI.-,r STAT(,*TSrnL) ..DIrMEMSTOIN 7(;53,47),TSOL (53,47)
C CALCUILATE 'PTAS A!D rnnT UEANI SOUARE ERROR ALSO THE MEAN OF ZFACT - 51 *. '45ZflFAN = a.RTAS . 'RrSr' o.DO i0 -=2,952 .:$'; f- ·rs- 3:X0C f :;t;f~f ,; ;,S0 :n O 10 J=> 46 - ' ' :RZMEA = 7r'EA.! -+ 7.(Tj) / FACT7 P = Z(T'.J) - TSnL(T,J)D2 =D*PR11S = RA +. n/FACTr.".,SF = RSE +. .D/FACT
10 COr-!'"T ,Jl F: r 's = SORT(RMSF) ' -P PTRr.T 20,7.FAf',PIASRigSE L. ./ -TA Z
2n FOR.5Y.AT At4HSTt.TIRTTCS OF SOLUTZDN,/lIO ,fEAMN Z=ttFl0.3,3X2IO NPTAS=' E12135X , 2Pi;E= .El2.3)RE TURN. :
EFFECi*JANFM.:. 1fNI) NfOOTIMIZE LTYECOU"IT(t60) ST7EftAx) AqJTODRL(NOPJE)
EFFEC *SOl!RCE..E.r!CDTIC 'OLIST m0tODFCK OBJECT MO40MAP 1'OFOPMAT 0GOST'T MtxRFrF
EFFEC,* FUICTIONS I NLINE ARe: NON.
EFFEC .*.
'F
.,OURCE STATFrTEPTS
NO .UTAGNOSTICS CGENERATED
COMrILATION ******
19' PROr)GRAM SIZE
ff' ' ' '$ t.S 0 0 -e f .,: . , 0 0 f: f f f 0 0 .~~~~~~~~~~~~~~~~
75T, SUPOROGRAM N
70K PYTES 0
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S-OU CE-Epc rIc rNOLT T ,,nr)ECKI ORJECT 0"JIt!A NO FOF MAT-GO..TfT ' OXPTF
FUNCTIONS INLTE An'E. NONEI._
TSN OOG2-isrI 0003
I SN 0006*I~- 0007ISM 0007T Ysri 0009
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TSN o0019ISN 002n-;:_ -' TSP! 0021
.,C, ISiH 0022TSJ 0023
D 0 - : I S-J 0 024 ISN 0025IS. 0026
'stirRnOJTT.IE [TIFOlJTi(TqOL,7,i.S:CAM): ;DI~HrgSTON T.~OL(5,~ttl7),7(53,~.7),rlIF(;~?47)
QVI'-;.^S'TOM KTLI'.~3).,'TARq9)l ,COr{q4),KrH§M(4::)PATA r.iTAp/-1 ,53,47,4?7,47,1tO953/DATA COP'/O.,1.v60o.,O./DA A KT[L/uH ,.'T.MPL,' HLM*,IHLTZ.,' EQ ',lSOL .,27*#H DATA fe, lrl/v- ST', ;2 P.Ds,$$SCA,qv v!,/DATA K TSO: /'T :L/n p.,r z, 71* Z,'DATA KnTF/'"IF ' i- :KT.(9): I='MrlU¢T ISCAN)iKTL(1O) = KUtM(3)FKTL(11) = KNUM14)i -- : : DO 1 I=l1,3
' nO 1 J=1,47 DIF(IJ) = t(IJ):- TSOL(IjJ)
1 CONTINUECC PRINT OUT SOLUTTION- ERROR FIELn . :' -: C
KTL(13) := TSnL . : CALl GRDPPT (TSOLNTArnCO.!,KTL 1,1) FTL¢13)-= KZ
CALL GRnmPT7fFZ HTASC,',KTL,1, : -.; -:KTL[1) : ~DIF ' . -:CALl.. GP PPRT(DIF,AITAR,CON,KTL , ,1)RETURNErND
----OPTIONS IN" EFFEC '*NAFE(MAT N ) MOOPTTM¶IZE LTNFCOUNT(6n) 'STZE(MAX) A'lTOOBLfLtlMNF)'CO _'O'TIONS IN EFFECs*SnuRcE ERCTTC r:OLIST !.InDECK OEJECT NOM/P MOFnRMAT GOSTMT' NOXRFF..
O= t TIONS IN EFFEC.* FUCTI)NtS TNLINE ARF_: -,O'JEJ*O1)TIOP!S IN EFFEC,* -.-*STATISTICS* ,UOURCE STATEMENT. = 5 , PPOGRA,¶ SIZE = 20882e SUPDROGRAW N/
'*.STATISTICS. NO. : AGN.!OSTICS GENERATED*:***** ErND OF CO'-ILATTON****** ' ' " 6.K RYTES OF
, .! . d~~~~~I
. .
SST MAP CLIMAT
Fig. 11. October average sea surface temperatures 0 K: Northern He.misphere.
The 270K isotherm approximates the ice edge.
-------T--,--,---. .. . .. d .. , ^ f - V . C ... . , , , ......
SST HAP CLIMAT
Fig. 12. November average sea surface temperatures OK: Northern HemiL
The 270K isotherm approximates the ice edge.
SST AP CLINAT
Fig. 13. December average sea surface temperatures K:The 270K isotherm approximates the ice edge.
Northern Hemisphere.
SST MAP CLIMAT
Fig. 14. January average sea surface temperatures oK: SouthernThe 270K isotherm approximates the ice edge.
-emisphere.Hemisph~ere.;0
SST MAP CLtIMAT
Fig. 15. February average sea surface temperatures K: Southern HemisphereThe 270K isotherm approximates the ice edge. ·
. .... ... .. ... ..
-, /0,,,,
0 I 0
- - .... ................ MAR WMCtINMC WASHINGTO
SST MAP CLIMAT
Fig. 16. March average sea surface temperatures OK: Southern HemisphEThe 270K isotherm approximates the ice edge.
SST MAP CLIMAT
Fig. 17: April average sea surface temperatures K: Southern Hemisphere.The 270K isotherm approximates the ice edge.
/.:.. ..... .... I - -
SST MAP CLIHAT
.g. 18. May average sea surface temperatures 0K: Southern Hemisphere.The 270K isotherm approximates the ice edge.
Fi
1, o :� � �
SST MAP CLIHAT
Fig. 19. J-ne average sea surface temperatures OK: Southern Hemisphere.
The 270K isotherm approximates the ice edge.
· 0
0~~~~~~~~~~~~~~1
y~~~~~~~~~~~~~~~~~~~~ eThe. 2
2/
9
0. -. ~~~~~~~~~~~~~~~~~~~~~~~.
0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ·~
SST HlAP CLIi'IAT
* * Fig. 20. July average sea surface temperatures OK: Southern Hemisphere.The 270K isotherm approximates. the ice edge .__
SST MAP CLIMAT
0Fig. 21. August average sea surface temperatures K: Southern Hemisphere.
The 270K isotherm approximates the ice edge. · - - E-- - X L - -.- · . ..Df
g :: : \ f X :: :~~~~ .. \
~~.0 Ct~e: u f : : a~f: D 7 f ::: f 0:: 0 0 : \?:: :. ·.a.
1~~ w
SST MAP CLIMAT
~. 22. September average sea surface temperatures K- Southern Hemispheres.'
The 270K isotherm approximates the ice edge., aFig
- i'01-. � � �, �,.: 7 �; i
SST 'MAP rCL,!ATL ,I: ..
Fig. 23. October average sea surface temperatures oK: Southern Hemisphere.
·;: :The 270K isotherm approximates the ice edge.
n ,,, _ Q | .7 , R S , , , , A ~ ~ ~~~~ , ., X,:!! .: ·., .:::: :i !jJ.i : -! : .- S :XX .
SST HAP CLIMAT
Fig. 24. November average sea surface temperatures K: Southern Hemisphere.
The 270K isotherm approximates the ice edge. _:
, .:� �� : � I , i
� I : � .� �, � I �.�
� I �- ��,�L �; !.~ -:: I t ,. '~ Ti': ff lf- .1''0 0 .'7 ' S'.S L i '' a:, ['d! I In' t., .0 'E: I
;00
SST HAP CLIMAT
Fig. 25. December average sea surface temperatures OK: Southern
The 270K isotherm approximates the ice edge.
Hemisphere