C FISHPAK3 FROM PORTLIB 12/30/83 SUBROUTINE HWSPLR (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, 1 ELMBDA,F,IDIMF,PERTRB,IERROR,W) C 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 3.1 , OCTOBER 1980) * 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 * * * * * * * * * PURPOSE * * * * * * * * * * * * * * * * * * C C C SUBROUTINE HWSPLR SOLVES A FINITE DIFFERENCE APPROXIMATION TO THE C HELMHOLTZ EQUATION IN POLAR COORDINATES: C C (1/R)(D/DR)(R(DU/DR)) + (1/R**2)(D/DTHETA)(DU/DTHETA) C C + LAMBDA*U = F(R,THETA). C C C C C * * * * * * * * PARAMETER DESCRIPTION * * * * * * * * * * C C * * * * * * ON INPUT * * * * * * C C A,B C THE RANGE OF R, I.E., A .LE. R .LE. B. A MUST BE LESS THAN B C AND A MUST BE NON-NEGATIVE. C C M C THE NUMBER OF PANELS INTO WHICH THE INTERVAL (A,B) IS C SUBDIVIDED. HENCE, THERE WILL BE M+1 GRID POINTS IN THE C R-DIRECTION GIVEN BY R(I) = A+(I-1)DR, FOR I = 1,2,...,M+1, C WHERE DR = (B-A)/M IS THE PANEL WIDTH. M MUST BE GREATER THAN 3. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITION AT R = A AND R = B. C C = 1 IF THE SOLUTION IS SPECIFIED AT R = A AND R = B. C = 2 IF THE SOLUTION IS SPECIFIED AT R = A AND THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO R IS SPECIFIED AT R = B. C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO R IS C SPECIFIED AT R = A (SEE NOTE BELOW) AND R = B. C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO R IS C SPECIFIED AT R = A (SEE NOTE BELOW) AND THE SOLUTION IS C SPECIFIED AT R = B. C = 5 IF THE SOLUTION IS UNSPECIFIED AT R = A = 0 AND THE C SOLUTION IS SPECIFIED AT R = B. C = 6 IF THE SOLUTION IS UNSPECIFIED AT R = A = 0 AND THE C DERIVATIVE OF THE SOLUTION WITH RESPECT TO R IS SPECIFIED C AT R = B. C C NOTE: IF A = 0, DO NOT USE MBDCND = 3 OR 4, BUT INSTEAD USE C MBDCND = 1,2,5, OR 6 . C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO R AT R = A. C WHEN MBDCND = 3 OR 4, C C BDA(J) = (D/DR)U(A,THETA(J)), J = 1,2,...,N+1 . C C WHEN MBDCND HAS ANY OTHER VALUE, BDA IS A DUMMY VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO R AT R = B. C WHEN MBDCND = 2,3, OR 6, C C BDB(J) = (D/DR)U(B,THETA(J)), J = 1,2,...,N+1 . C C WHEN MBDCND HAS ANY OTHER VALUE, BDB IS A DUMMY VARIABLE. C C C,D C THE RANGE OF THETA, I.E., C .LE. THETA .LE. D. C MUST BE LESS C THAN D. C C N C THE NUMBER OF PANELS INTO WHICH THE INTERVAL (C,D) IS C SUBDIVIDED. HENCE, THERE WILL BE N+1 GRID POINTS IN THE C THETA-DIRECTION GIVEN BY THETA(J) = C+(J-1)DTHETA FOR C J = 1,2,...,N+1, WHERE DTHETA = (D-C)/N IS THE PANEL WIDTH. N C MUST BE GREATER THAN 3. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT THETA = C AND C AND THETA = D. C C = 0 IF THE SOLUTION IS PERIODIC IN THETA, I.E., C U(I,J) = U(I,N+J). C = 1 IF THE SOLUTION IS SPECIFIED AT THETA = C AND THETA = D C (SEE NOTE BELOW). C = 2 IF THE SOLUTION IS SPECIFIED AT THETA = C AND THE C DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = D (SEE NOTE BELOW). C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = C AND THE SOLUTION IS SPECIFIED AT C THETA = D (SEE NOTE BELOW). C C NOTE: WHEN NBDCND = 1,2, OR 4, DO NOT USE MBDCND = 5 OR 6 C (THE FORMER INDICATES THAT THE SOLUTION IS SPECIFIED AT C R = 0, THE LATTER INDICATES THE SOLUTION IS UNSPECIFIED C AT R = 0). USE INSTEAD MBDCND = 1 OR 2 . C C BDC C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA AT C THETA = C. WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DTHETA)U(R(I),C), I = 1,2,...,M+1 . C C WHEN NBDCND HAS ANY OTHER VALUE, BDC IS A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO THETA AT C THETA = D. WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DTHETA)U(R(I),D), I = 1,2,...,M+1 . C C WHEN NBDCND HAS ANY OTHER VALUE, BDD IS A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE HELMHOLTZ EQUATION. IF C LAMBDA .LT. 0, A SOLUTION MAY NOT EXIST. HOWEVER, HWSPLR WILL C ATTEMPT TO FIND A SOLUTION. C C F C A TWO-DIMENSIONAL ARRAY THAT SPECIFIES THE VALUES OF THE RIGHT C SIDE OF THE HELMHOLTZ EQUATION AND BOUNDARY VALUES (IF ANY). C FOR I = 2,3,...,M AND J = 2,3,...,N C C F(I,J) = F(R(I),THETA(J)). C C ON THE BOUNDARIES F IS DEFINED BY C C MBDCND F(1,J) F(M+1,J) C ------ ------------- ------------- C C 1 U(A,THETA(J)) U(B,THETA(J)) C 2 U(A,THETA(J)) F(B,THETA(J)) C 3 F(A,THETA(J)) F(B,THETA(J)) C 4 F(A,THETA(J)) U(B,THETA(J)) J = 1,2,...,N+1 C 5 F(0,0) U(B,THETA(J)) C 6 F(0,0) F(B,THETA(J)) C C NBDCND F(I,1) F(I,N+1) C ------ --------- --------- C C 0 F(R(I),C) F(R(I),C) C 1 U(R(I),C) U(R(I),D) C 2 U(R(I),C) F(R(I),D) I = 1,2,...,M+1 C 3 F(R(I),C) F(R(I),D) C 4 F(R(I),C) U(R(I),D) C C F MUST BE DIMENSIONED AT LEAST (M+1)*(N+1). C C NOTE C C IF THE TABLE CALLS FOR BOTH THE SOLUTION U AND THE RIGHT SIDE F C AT A CORNER THEN THE SOLUTION MUST BE SPECIFIED. C C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY F AS IT APPEARS IN THE C PROGRAM CALLING HWSPLR. THIS PARAMETER IS USED TO SPECIFY THE C VARIABLE DIMENSION OF F. IDIMF MUST BE AT LEAST M+1 . C C W C A ONE-DIMENSIONAL ARRAY THAT MUST BE PROVIDED BY THE USER FOR C WORK SPACE. W MAY REQUIRE UP TO 4*(N+1) + C (13 + INT(LOG2(N+1)))*(M+1) LOCATIONS. THE ACTUAL NUMBER OF C LOCATIONS USED IS COMPUTED BY HWSPLR AND IS RETURNED IN LOCATION C W(1). C C C * * * * * * ON OUTPUT * * * * * * C C F C CONTAINS THE SOLUTION U(I,J) OF THE FINITE DIFFERENCE C APPROXIMATION FOR THE GRID POINT (R(I),THETA(J)), C I = 1,2,...,M+1, J = 1,2,...,N+1 . C C PERTRB C IF A COMBINATION OF PERIODIC, DERIVATIVE, OR UNSPECIFIED C BOUNDARY CONDITIONS IS SPECIFIED FOR A POISSON EQUATION C (LAMBDA = 0), A SOLUTION MAY NOT EXIST. PERTRB IS A CONSTANT, C CALCULATED AND SUBTRACTED FROM F, WHICH ENSURES THAT A SOLUTION C EXISTS. HWSPLR THEN COMPUTES THIS SOLUTION, WHICH IS A LEAST C SQUARES SOLUTION TO THE ORIGINAL APPROXIMATION. THIS SOLUTION C PLUS ANY CONSTANT IS ALSO A SOLUTION. HENCE, THE SOLUTION IS C NOT UNIQUE. PERTRB SHOULD BE SMALL COMPARED TO THE RIGHT SIDE. C OTHERWISE, A SOLUTION IS OBTAINED TO AN ESSENTIALLY DIFFERENT C PROBLEM. THIS COMPARISON SHOULD ALWAYS BE MADE TO INSURE THAT A C MEANINGFUL SOLUTIN HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. EXCEPT C FOR NUMBERS 0 AND 11, A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR. C = 1 A .LT. 0 . C = 2 A .GE. B. C = 3 MBDCND .LT. 1 OR MBDCND .GT. 6 . C = 4 C .GE. D. C = 5 N .LE. 3 C = 6 NBDCND .LT. 0 OR .GT. 4 . C = 7 A = 0, MBDCND = 3 OR 4 . C = 8 A .GT. 0, MBDCND .GE. 5 . C = 9 MBDCND .GE. 5, NBDCND .NE. 0 AND NBDCND .NE. 3 . C = 10 IDIMF .LT. M+1 . C = 11 LAMBDA .GT. 0 . C = 12 M .LE. 3 C C SINCE THIS IS THE ONLY MEANS OF INDICATING A POSSIBLY INCORRECT C CALL TO HWSPLR, THE USER SHOULD TEST IERROR AFTER THE CALL. C C W C W(1) CONTAINS THE REQUIRED LENGTH OF W. C C C * * * * * * * PROGRAM SPECIFICATIONS * * * * * * * * * * * * C C DIMENSION OF BDA(N+1),BDB(N+1),BDC(M+1),BDD(M+1),F(IDIMF,N+1), C ARGUMENTS W(SEE ARGUMENT LIST) C C LATEST JUNE 1, 1976 C REVISION C C SUBPROGRAMS HWSPLR,GENBUN,POISD2,POISN2,POISP2,COSGEN,MERGE, C REQUIRED TRIX,TRI3,PIMACH C C SPECIAL NONE C CONDITIONS C C COMMON NONE C BLOCKS C C I/O C C PRECISION SINGLE C C SPECIALIST ROLAND SWEET C C LANGUAGE FORTRAN C C HISTORY STANDARDIZED APRIL 1, 1973 C REVISED JANUARY 1, 1976 C C ALGORITHM THE ROUTINE DEFINES THE FINITE DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, AND ADJUSTS C THE RIGHT SIDE OF SINGULAR SYSTEMS AND THEN CALLS C GENBUN TO SOLVE THE SYSTEM. C C SPACE 13430(OCTAL) = 5912(DECIMAL) LOCATIONS ON THE NCAR C REQUIRED CONTROL DATA 7600 C C TIMING AND THE EXECUTION TIME T ON THE NCAR CONTROL DATA C ACCURACY 7600 FOR SUBROUTINE HWSPLR IS ROUGHLY PROPORTIONAL C TO M*N*LOG2(N), BUT ALSO DEPENDS ON THE INPUT C PARAMETERS NBDCND AND MBDCND. SOME TYPICAL VALUES C ARE LISTED IN THE TABLE BELOW. C THE SOLUTION PROCESS EMPLOYED RESULTS IN A LOSS C OF NO MORE THAN THREE SIGNIFICANT DIGITS FOR N AND C M AS LARGE AS 64. MORE DETAILED INFORMATION 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 C M(=N) MBDCND NBDCND T(MSECS) C ----- ------ ------ -------- C C 32 1 0 31 C 32 1 1 23 C 32 3 3 36 C 64 1 0 128 C 64 1 1 96 C 64 3 3 142 C C PORTABILITY AMERICAN NATIONAL STANDARDS INSTITUTE FORTRAN. C ALL MACHINE DEPENDENT CONSTANTS ARE LOCATED IN THE C FUNCTION PIMACH. C C REQUIRED COS C RESIDENT C ROUTINES C C REFERENCE SWARZTRAUBER,P. AND R. SWEET, 'EFFICIENT FORTRAN C SUBPROGRAMS FOR THE SOLUTION OF ELLIPTIC EQUATIONS' C NCAR TN/IA-109, JULY, 1975, 138 PP. C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C DIMENSION F(IDIMF,1) DIMENSION BDA(1) ,BDB(1) ,BDC(1) ,BDD(1) , 1 W(1) C C CHECK FOR INVALID PARAMETERS. C IERROR = 0 IF (A .LT. 0.) IERROR = 1 IF (A .GE. B) IERROR = 2 IF (MBDCND.LE.0 .OR. MBDCND.GE.7) IERROR = 3 IF (C .GE. D) IERROR = 4 IF (N .LE. 3) IERROR = 5 IF (NBDCND.LE.-1 .OR. NBDCND.GE.5) IERROR = 6 IF (A.EQ.0. .AND. (MBDCND.EQ.3 .OR. MBDCND.EQ.4)) IERROR = 7 IF (A.GT.0. .AND. MBDCND.GE.5) IERROR = 8 IF (MBDCND.GE.5 .AND. NBDCND.NE.0 .AND. NBDCND.NE.3) IERROR = 9 IF (IDIMF .LT. M+1) IERROR = 10 IF (M .LE. 3) IERROR = 12 IF (IERROR .NE. 0) RETURN MP1 = M+1 DELTAR = (B-A)/FLOAT(M) DLRBY2 = DELTAR/2. DLRSQ = DELTAR**2 NP1 = N+1 DELTHT = (D-C)/FLOAT(N) DLTHSQ = DELTHT**2 NP = NBDCND+1 C C DEFINE RANGE OF INDICES I AND J FOR UNKNOWNS U(I,J). C MSTART = 2 MSTOP = MP1 GO TO (101,105,102,103,104,105),MBDCND 101 MSTOP = M GO TO 105 102 MSTART = 1 GO TO 105 103 MSTART = 1 104 MSTOP = M 105 MUNK = MSTOP-MSTART+1 NSTART = 1 NSTOP = N GO TO (109,106,107,108,109),NP 106 NSTART = 2 GO TO 109 107 NSTART = 2 108 NSTOP = NP1 109 NUNK = NSTOP-NSTART+1 C C DEFINE A,B,C COEFFICIENTS IN W-ARRAY. C ID2 = MUNK ID3 = ID2+MUNK ID4 = ID3+MUNK ID5 = ID4+MUNK ID6 = ID5+MUNK A1 = 2./DLRSQ IJ = 0 IF (MBDCND.EQ.3 .OR. MBDCND.EQ.4) IJ = 1 DO 110 I=1,MUNK R = A+FLOAT(I-IJ)*DELTAR J = ID5+I W(J) = R J = ID6+I W(J) = 1./R**2 W(I) = (R-DLRBY2)/(R*DLRSQ) J = ID3+I W(J) = (R+DLRBY2)/(R*DLRSQ) J = ID2+I W(J) = -A1+ELMBDA 110 CONTINUE GO TO (114,111,112,113,114,111),MBDCND 111 W(ID2) = A1 GO TO 114 112 W(ID2) = A1 113 W(ID3+1) = A1 114 CONTINUE C C ENTER BOUNDARY DATA FOR R-BOUNDARIES. C GO TO (115,115,117,117,119,119),MBDCND 115 A1 = W(1) DO 116 J=NSTART,NSTOP F(2,J) = F(2,J)-A1*F(1,J) 116 CONTINUE GO TO 119 117 A1 = 2.*DELTAR*W(1) DO 118 J=NSTART,NSTOP F(1,J) = F(1,J)+A1*BDA(J) 118 CONTINUE 119 GO TO (120,122,122,120,120,122),MBDCND 120 A1 = W(ID4) DO 121 J=NSTART,NSTOP F(M,J) = F(M,J)-A1*F(MP1,J) 121 CONTINUE GO TO 124 122 A1 = 2.*DELTAR*W(ID4) DO 123 J=NSTART,NSTOP F(MP1,J) = F(MP1,J)-A1*BDB(J) 123 CONTINUE C C ENTER BOUNDARY DATA FOR THETA-BOUNDARIES. C 124 A1 = 1./DLTHSQ L = ID5-MSTART+1 LP = ID6-MSTART+1 GO TO (134,125,125,127,127),NP 125 DO 126 I=MSTART,MSTOP J = I+LP F(I,2) = F(I,2)-A1*W(J)*F(I,1) 126 CONTINUE GO TO 129 127 A1 = 2./DELTHT DO 128 I=MSTART,MSTOP J = I+LP F(I,1) = F(I,1)+A1*W(J)*BDC(I) 128 CONTINUE 129 A1 = 1./DLTHSQ GO TO (134,130,132,132,130),NP 130 DO 131 I=MSTART,MSTOP J = I+LP F(I,N) = F(I,N)-A1*W(J)*F(I,NP1) 131 CONTINUE GO TO 134 132 A1 = 2./DELTHT DO 133 I=MSTART,MSTOP J = I+LP F(I,NP1) = F(I,NP1)-A1*W(J)*BDD(I) 133 CONTINUE 134 CONTINUE C C ADJUST RIGHT SIDE OF EQUATION FOR UNKNOWN AT POLE WHEN HAVE C DERIVATIVE SPECIFIED BOUNDARY CONDITIONS. C IF (MBDCND.GE.5 .AND. NBDCND.EQ.3) 1 F(1,1) = F(1,1)-(BDD(2)-BDC(2))*4./(FLOAT(N)*DELTHT*DLRSQ) C C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A C SOLUTION. C PERTRB = 0. IF (ELMBDA) 144,136,135 135 IERROR = 11 GO TO 144 136 IF (NBDCND.NE.0 .AND. NBDCND.NE.3) GO TO 144 S2 = 0. GO TO (144,144,137,144,144,138),MBDCND 137 W(ID5+1) = .5*(W(ID5+2)-DLRBY2) S2 = .25*DELTAR 138 A2 = 2. IF (NBDCND .EQ. 0) A2 = 1. J = ID5+MUNK W(J) = .5*(W(J-1)+DLRBY2) S = 0. DO 140 I=MSTART,MSTOP S1 = 0. IJ = NSTART+1 K = NSTOP-1 DO 139 J=IJ,K S1 = S1+F(I,J) 139 CONTINUE J = I+L S = S+(A2*S1+F(I,NSTART)+F(I,NSTOP))*W(J) 140 CONTINUE S2 = FLOAT(M)*A+DELTAR*(FLOAT((M-1)*(M+1))*.5+.25)+S2 S1 = (2.+A2*FLOAT(NUNK-2))*S2 IF (MBDCND .EQ. 3) GO TO 141 S2 = FLOAT(N)*A2*DELTAR/8. S = S+F(1,1)*S2 S1 = S1+S2 141 CONTINUE PERTRB = S/S1 DO 143 I=MSTART,MSTOP DO 142 J=NSTART,NSTOP F(I,J) = F(I,J)-PERTRB 142 CONTINUE 143 CONTINUE 144 CONTINUE C C MULTIPLY I-TH EQUATION THROUGH BY (R(I)*DELTHT)**2. C DO 146 I=MSTART,MSTOP K = I-MSTART+1 J = I+LP A1 = DLTHSQ/W(J) W(K) = A1*W(K) J = ID2+K W(J) = A1*W(J) J = ID3+K W(J) = A1*W(J) DO 145 J=NSTART,NSTOP F(I,J) = A1*F(I,J) 145 CONTINUE 146 CONTINUE W(1) = 0. W(ID4) = 0. C C CALL GENBUN TO SOLVE THE SYSTEM OF EQUATIONS. C CALL GENBUN (NBDCND,NUNK,1,MUNK,W(1),W(ID2+1),W(ID3+1),IDIMF, 1 F(MSTART,NSTART),IERR1,W(ID4+1)) IWSTOR = W(ID4+1)+3.*FLOAT(MUNK) GO TO (157,157,157,157,148,147),MBDCND C C ADJUST THE SOLUTION AS NECESSARY FOR THE PROBLEMS WHERE A = 0. C 147 IF (ELMBDA .NE. 0.) GO TO 148 YPOLE = 0. GO TO 155 148 CONTINUE J = ID5+MUNK W(J) = W(ID2)/W(ID3) DO 149 IP=3,MUNK I = MUNK-IP+2 J = ID5+I LP = ID2+I K = ID3+I W(J) = W(I)/(W(LP)-W(K)*W(J+1)) 149 CONTINUE W(ID5+1) = -.5*DLTHSQ/(W(ID2+1)-W(ID3+1)*W(ID5+2)) DO 150 I=2,MUNK J = ID5+I W(J) = -W(J)*W(J-1) 150 CONTINUE S = 0. DO 151 J=NSTART,NSTOP S = S+F(2,J) 151 CONTINUE A2 = NUNK IF (NBDCND .EQ. 0) GO TO 152 S = S-.5*(F(2,NSTART)+F(2,NSTOP)) A2 = A2-1. 152 YPOLE = (.25*DLRSQ*F(1,1)-S/A2)/(W(ID5+1)-1.+ELMBDA*DLRSQ*.25) DO 154 I=MSTART,MSTOP K = L+I DO 153 J=NSTART,NSTOP F(I,J) = F(I,J)+YPOLE*W(K) 153 CONTINUE 154 CONTINUE 155 DO 156 J=1,NP1 F(1,J) = YPOLE 156 CONTINUE 157 CONTINUE IF (NBDCND .NE. 0) GO TO 159 DO 158 I=MSTART,MSTOP F(I,NP1) = F(I,1) 158 CONTINUE 159 CONTINUE W(1) = IWSTOR RETURN END