SUBROUTINE HWSCRT (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, 1 ELMBDA,F,IDIMF,PERTRB,IERROR,W) C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * * C * F I S H P A K * C * * C * * C * A PACKAGE OF FORTRAN SUBPROGRAMS FOR THE SOLUTION OF * C * * C * SEPARABLE ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS * C * * C * (VERSION 3.2 , NOVEMBER 1988) * C * * C * BY * C * * C * JOHN ADAMS, PAUL SWARZTRAUBER AND ROLAND SWEET * C * * C * OF * C * * C * THE NATIONAL CENTER FOR ATMOSPHERIC RESEARCH * C * * C * BOULDER, COLORADO (80307) U.S.A. * C * * C * WHICH IS SPONSORED BY * C * * C * THE NATIONAL SCIENCE FOUNDATION * C * * C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C C C DIMENSION OF BDA(N), BDB(N), BDC(M),BDD(M), C ARGUMENTS F(IDIMF,N), W(SEE ARGUMENT LIST) C C LATEST REVISION NOVEMBER 1988 C C PURPOSE SOLVES THE STANDARD FIVE-POINT FINITE C DIFFERENCE APPROXIMATION TO THE HELMHOLTZ C EQUATION IN CARTESIAN COORDINATES. THIS C EQUATION IS C C (D/DX)(DU/DX) + (D/DY)(DU/DY) C + LAMBDA*U = F(X,Y). C C USAGE CALL HWSCRT (A,B,M,MBDCND,BDA,BDB,C,D,N, C NBDCND,BDC,BDD,ELMBDA,F,IDIMF, C PERTRB,IERROR,W) C C ARGUMENTS C ON INPUT A,B C C THE RANGE OF X, I.E., A .LE. X .LE. B. C A MUST BE LESS THAN B. C C M C THE NUMBER OF PANELS INTO WHICH THE C INTERVAL (A,B) IS SUBDIVIDED. C HENCE, THERE WILL BE M+1 GRID POINTS C IN THE X-DIRECTION GIVEN BY C X(I) = A+(I-1)DX FOR I = 1,2,...,M+1, C WHERE DX = (B-A)/M IS THE PANEL WIDTH. C M MUST BE GREATER THAN 3. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS C AT X = A AND X = B. C C = 0 IF THE SOLUTION IS PERIODIC IN X, C I.E., U(I,J) = U(M+I,J). C = 1 IF THE SOLUTION IS SPECIFIED AT C X = A AND X = B. C = 2 IF THE SOLUTION IS SPECIFIED AT C X = A AND THE DERIVATIVE OF THE C SOLUTION WITH RESPECT TO X IS C SPECIFIED AT X = B. C = 3 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO X IS SPECIFIED AT C AT X = A AND X = B. C = 4 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO X IS SPECIFIED AT C X = A AND THE SOLUTION IS SPECIFIED C AT X = B. C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO X AT X = A. C C WHEN MBDCND = 3 OR 4, C C BDA(J) = (D/DX)U(A,Y(J)), J = 1,2,...,N+1. C C WHEN MBDCND HAS ANY OTHER VALUE, BDA IS C A DUMMY VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 C THAT SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO X AT X = B. C C WHEN MBDCND = 2 OR 3, C C BDB(J) = (D/DX)U(B,Y(J)), J = 1,2,...,N+1 C C WHEN MBDCND HAS ANY OTHER VALUE BDB IS A C DUMMY VARIABLE. C C C,D C THE RANGE OF Y, I.E., C .LE. Y .LE. D. C C MUST BE LESS THAN D. C C N C THE NUMBER OF PANELS INTO WHICH THE C INTERVAL (C,D) IS SUBDIVIDED. HENCE, C THERE WILL BE N+1 GRID POINTS IN THE C Y-DIRECTION GIVEN BY Y(J) = C+(J-1)DY C FOR J = 1,2,...,N+1, WHERE C DY = (D-C)/N IS THE PANEL WIDTH. C N MUST BE GREATER THAN 3. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT C Y = C AND Y = D. C C = 0 IF THE SOLUTION IS PERIODIC IN Y, C I.E., U(I,J) = U(I,N+J). C = 1 IF THE SOLUTION IS SPECIFIED AT C Y = C AND Y = D. C = 2 IF THE SOLUTION IS SPECIFIED AT C Y = C AND THE DERIVATIVE OF THE C SOLUTION WITH RESPECT TO Y IS C SPECIFIED AT Y = D. C = 3 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO Y IS SPECIFIED AT C Y = C AND Y = D. C = 4 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO Y IS SPECIFIED AT C Y = C AND THE SOLUTION IS SPECIFIED C AT Y = D. C C BDC C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO Y AT Y = C. C C WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DY)U(X(I),C), I = 1,2,...,M+1 C C WHEN NBDCND HAS ANY OTHER VALUE, BDC IS C A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO Y AT Y = D. C C WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DY)U(X(I),D), I = 1,2,...,M+1 C C WHEN NBDCND HAS ANY OTHER VALUE, BDD IS C A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE HELMHOLTZ C EQUATION. IF LAMBDA .GT. 0, A SOLUTION C MAY NOT EXIST. HOWEVER, HWSCRT WILL C ATTEMPT TO FIND A SOLUTION. C C F C A TWO-DIMENSIONAL ARRAY, OF DIMENSION AT C LEAST (M+1)*(N+1), SPECIFYING VALUES OF THE C RIGHT SIDE OF THE HELMHOLTZ EQUATION AND C BOUNDARY VALUES (IF ANY). C C ON THE INTERIOR, F IS DEFINED AS FOLLOWS: C FOR I = 2,3,...,M AND J = 2,3,...,N C F(I,J) = F(X(I),Y(J)). C C ON THE BOUNDARIES, F IS DEFINED AS FOLLOWS: C FOR J=1,2,...,N+1, I=1,2,...,M+1, C C MBDCND F(1,J) F(M+1,J) C ------ --------- -------- C C 0 F(A,Y(J)) F(A,Y(J)) C 1 U(A,Y(J)) U(B,Y(J)) C 2 U(A,Y(J)) F(B,Y(J)) C 3 F(A,Y(J)) F(B,Y(J)) C 4 F(A,Y(J)) U(B,Y(J)) C C C NBDCND F(I,1) F(I,N+1) C ------ --------- -------- C C 0 F(X(I),C) F(X(I),C) C 1 U(X(I),C) U(X(I),D) C 2 U(X(I),C) F(X(I),D) C 3 F(X(I),C) F(X(I),D) C 4 F(X(I),C) U(X(I),D) C C NOTE: C IF THE TABLE CALLS FOR BOTH THE SOLUTION U C AND THE RIGHT SIDE F AT A CORNER THEN THE C SOLUTION MUST BE SPECIFIED. C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY C F AS IT APPEARS IN THE PROGRAM CALLING C HWSCRT. THIS PARAMETER IS USED TO SPECIFY C THE VARIABLE DIMENSION OF F. IDIMF MUST C BE AT LEAST M+1 . C C W C A ONE-DIMENSIONAL ARRAY THAT MUST BE C PROVIDED BY THE USER FOR WORK SPACE. C W MAY REQUIRE UP TO 4*(N+1) + C (13 + INT(LOG2(N+1)))*(M+1) LOCATIONS. C THE ACTUAL NUMBER OF LOCATIONS USED IS C COMPUTED BY HWSCRT AND IS RETURNED IN C LOCATION W(1). C C C ON OUTPUT F C CONTAINS THE SOLUTION U(I,J) OF THE FINITE C DIFFERENCE APPROXIMATION FOR THE GRID POINT C (X(I),Y(J)), I = 1,2,...,M+1, C J = 1,2,...,N+1 . C C PERTRB C IF A COMBINATION OF PERIODIC OR DERIVATIVE C BOUNDARY CONDITIONS IS SPECIFIED FOR A C POISSON EQUATION (LAMBDA = 0), A SOLUTION C MAY NOT EXIST. PERTRB IS A CONSTANT, C CALCULATED AND SUBTRACTED FROM F, WHICH C ENSURES THAT A SOLUTION EXISTS. HWSCRT C THEN COMPUTES THIS SOLUTION, WHICH IS A C LEAST SQUARES SOLUTION TO THE ORIGINAL C APPROXIMATION. THIS SOLUTION PLUS ANY C CONSTANT IS ALSO A SOLUTION. HENCE, THE C SOLUTION IS NOT UNIQUE. THE VALUE OF C PERTRB SHOULD BE SMALL COMPARED TO THE C RIGHT SIDE F. OTHERWISE, A SOLUTION IS C OBTAINED TO AN ESSENTIALLY DIFFERENT C PROBLEM. THIS COMPARISON SHOULD ALWAYS C BE MADE TO INSURE THAT A MEANINGFUL C SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT C PARAMETERS. EXCEPT FOR NUMBERS 0 AND 6, C A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR C = 1 A .GE. B C = 2 MBDCND .LT. 0 OR MBDCND .GT. 4 C = 3 C .GE. D C = 4 N .LE. 3 C = 5 NBDCND .LT. 0 OR NBDCND .GT. 4 C = 6 LAMBDA .GT. 0 C = 7 IDIMF .LT. M+1 C = 8 M .LE. 3 C C SINCE THIS IS THE ONLY MEANS OF INDICATING C A POSSIBLY INCORRECT CALL TO HWSCRT, THE C USER SHOULD TEST IERROR AFTER THE CALL. C C W C W(1) CONTAINS THE REQUIRED LENGTH OF W. C C SPECIAL CONDITIONS NONE C C I/O NONE C C PRECISION SINGLE C C REQUIRED LIBRARY GENBUN, GNBNAUX, AND COMF C FILES FROM FISHPAK C C LANGUAGE FORTRAN C C HISTORY WRITTEN BY ROLAND SWEET AT NCAR IN THE LATE C 1970'S. RELEASED ON NCAR'S PUBLIC SOFTWARE C LIBRARIES IN JANUARY 1980. C C PORTABILITY FORTRAN 77 C C ALGORITHM THE ROUTINE DEFINES THE FINITE DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, AND C ADJUSTS THE RIGHT SIDE OF SINGULAR SYSTEMS C AND THEN CALLS GENBUN TO SOLVE THE SYSTEM. C C TIMING FOR LARGE M AND N, THE OPERATION COUNT C IS ROUGHLY PROPORTIONAL TO C M*N*(LOG2(N) C BUT ALSO DEPENDS ON INPUT PARAMETERS NBDCND C AND MBDCND. C C ACCURACY THE SOLUTION PROCESS EMPLOYED RESULTS IN A LOSS C OF NO MORE THAN THREE SIGNIFICANT DIGITS FOR N C AND M AS LARGE AS 64. MORE DETAILS ABOUT C ACCURACY CAN BE FOUND IN THE DOCUMENTATION FOR C SUBROUTINE GENBUN WHICH IS THE ROUTINE THAT C SOLVES THE FINITE DIFFERENCE EQUATIONS. C C REFERENCES SWARZTRAUBER,P. AND R. SWEET, "EFFICIENT C FORTRAN SUBPROGRAMS FOR THE SOLUTION OF C ELLIPTIC EQUATIONS" C NCAR TN/IA-109, JULY, 1975, 138 PP. C*********************************************************************** DIMENSION F(IDIMF,1) DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) , 1 W(*) C C CHECK FOR INVALID PARAMETERS. C IERROR = 0 IF (A .GE. B) IERROR = 1 IF (MBDCND.LT.0 .OR. MBDCND.GT.4) IERROR = 2 IF (C .GE. D) IERROR = 3 IF (N .LE. 3) IERROR = 4 IF (NBDCND.LT.0 .OR. NBDCND.GT.4) IERROR = 5 IF (IDIMF .LT. M+1) IERROR = 7 IF (M .LE. 3) IERROR = 8 IF (IERROR .NE. 0) RETURN NPEROD = NBDCND MPEROD = 0 IF (MBDCND .GT. 0) MPEROD = 1 DELTAX = (B-A)/FLOAT(M) TWDELX = 2./DELTAX DELXSQ = 1./DELTAX**2 DELTAY = (D-C)/FLOAT(N) TWDELY = 2./DELTAY DELYSQ = 1./DELTAY**2 NP = NBDCND+1 NP1 = N+1 MP = MBDCND+1 MP1 = M+1 NSTART = 1 NSTOP = N NSKIP = 1 GO TO (104,101,102,103,104),NP 101 NSTART = 2 GO TO 104 102 NSTART = 2 103 NSTOP = NP1 NSKIP = 2 104 NUNK = NSTOP-NSTART+1 C C ENTER BOUNDARY DATA FOR X-BOUNDARIES. C MSTART = 1 MSTOP = M MSKIP = 1 GO TO (117,105,106,109,110),MP 105 MSTART = 2 GO TO 107 106 MSTART = 2 MSTOP = MP1 MSKIP = 2 107 DO 108 J=NSTART,NSTOP F(2,J) = F(2,J)-F(1,J)*DELXSQ 108 CONTINUE GO TO 112 109 MSTOP = MP1 MSKIP = 2 110 DO 111 J=NSTART,NSTOP F(1,J) = F(1,J)+BDA(J)*TWDELX 111 CONTINUE 112 GO TO (113,115),MSKIP 113 DO 114 J=NSTART,NSTOP F(M,J) = F(M,J)-F(MP1,J)*DELXSQ 114 CONTINUE GO TO 117 115 DO 116 J=NSTART,NSTOP F(MP1,J) = F(MP1,J)-BDB(J)*TWDELX 116 CONTINUE 117 MUNK = MSTOP-MSTART+1 C C ENTER BOUNDARY DATA FOR Y-BOUNDARIES. C GO TO (127,118,118,120,120),NP 118 DO 119 I=MSTART,MSTOP F(I,2) = F(I,2)-F(I,1)*DELYSQ 119 CONTINUE GO TO 122 120 DO 121 I=MSTART,MSTOP F(I,1) = F(I,1)+BDC(I)*TWDELY 121 CONTINUE 122 GO TO (123,125),NSKIP 123 DO 124 I=MSTART,MSTOP F(I,N) = F(I,N)-F(I,NP1)*DELYSQ 124 CONTINUE GO TO 127 125 DO 126 I=MSTART,MSTOP F(I,NP1) = F(I,NP1)-BDD(I)*TWDELY 126 CONTINUE C C MULTIPLY RIGHT SIDE BY DELTAY**2. C 127 DELYSQ = DELTAY*DELTAY DO 129 I=MSTART,MSTOP DO 128 J=NSTART,NSTOP F(I,J) = F(I,J)*DELYSQ 128 CONTINUE 129 CONTINUE C C DEFINE THE A,B,C COEFFICIENTS IN W-ARRAY. C ID2 = MUNK ID3 = ID2+MUNK ID4 = ID3+MUNK S = DELYSQ*DELXSQ ST2 = 2.*S DO 130 I=1,MUNK W(I) = S J = ID2+I W(J) = -ST2+ELMBDA*DELYSQ J = ID3+I W(J) = S 130 CONTINUE IF (MP .EQ. 1) GO TO 131 W(1) = 0. W(ID4) = 0. 131 CONTINUE GO TO (135,135,132,133,134),MP 132 W(ID2) = ST2 GO TO 135 133 W(ID2) = ST2 134 W(ID3+1) = ST2 135 CONTINUE PERTRB = 0. IF (ELMBDA) 144,137,136 136 IERROR = 6 GO TO 144 137 IF ((NBDCND.EQ.0 .OR. NBDCND.EQ.3) .AND. 1 (MBDCND.EQ.0 .OR. MBDCND.EQ.3)) GO TO 138 GO TO 144 C C FOR SINGULAR PROBLEMS MUST ADJUST DATA TO INSURE THAT A SOLUTION C WILL EXIST. C 138 A1 = 1. A2 = 1. IF (NBDCND .EQ. 3) A2 = 2. IF (MBDCND .EQ. 3) A1 = 2. S1 = 0. MSP1 = MSTART+1 MSTM1 = MSTOP-1 NSP1 = NSTART+1 NSTM1 = NSTOP-1 DO 140 J=NSP1,NSTM1 S = 0. DO 139 I=MSP1,MSTM1 S = S+F(I,J) 139 CONTINUE S1 = S1+S*A1+F(MSTART,J)+F(MSTOP,J) 140 CONTINUE S1 = A2*S1 S = 0. DO 141 I=MSP1,MSTM1 S = S+F(I,NSTART)+F(I,NSTOP) 141 CONTINUE S1 = S1+S*A1+F(MSTART,NSTART)+F(MSTART,NSTOP)+F(MSTOP,NSTART)+ 1 F(MSTOP,NSTOP) S = (2.+FLOAT(NUNK-2)*A2)*(2.+FLOAT(MUNK-2)*A1) PERTRB = S1/S DO 143 J=NSTART,NSTOP DO 142 I=MSTART,MSTOP F(I,J) = F(I,J)-PERTRB 142 CONTINUE 143 CONTINUE PERTRB = PERTRB/DELYSQ C C SOLVE THE EQUATION. C 144 CALL GENBUN (NPEROD,NUNK,MPEROD,MUNK,W(1),W(ID2+1),W(ID3+1), 1 IDIMF,F(MSTART,NSTART),IERR1,W(ID4+1)) W(1) = W(ID4+1)+3.*FLOAT(MUNK) C C FILL IN IDENTICAL VALUES WHEN HAVE PERIODIC BOUNDARY CONDITIONS. C IF (NBDCND .NE. 0) GO TO 146 DO 145 I=MSTART,MSTOP F(I,NP1) = F(I,1) 145 CONTINUE 146 IF (MBDCND .NE. 0) GO TO 148 DO 147 J=NSTART,NSTOP F(MP1,J) = F(1,J) 147 CONTINUE IF (NBDCND .EQ. 0) F(MP1,NP1) = F(1,NP1) 148 CONTINUE RETURN C C REVISION HISTORY--- C C SEPTEMBER 1973 VERSION 1 C APRIL 1976 VERSION 2 C JANUARY 1978 VERSION 3 C DECEMBER 1979 VERSION 3.1 C FEBRUARY 1985 DOCUMENTATION UPGRADE C NOVEMBER 1988 VERSION 3.2, FORTRAN 77 CHANGES C----------------------------------------------------------------------- END