C FISHPAK9 FROM PORTLIB 12/30/83 SUBROUTINE HSTCYL (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 HSTCYL SOLVES THE STANDARD FIVE-POINT FINITE DIFFERENCE C APPROXIMATION ON A STAGGERED GRID TO THE MODIFIED HELMHOLTZ C EQUATION IN CYLINDRICAL COORDINATES C C (1/R)(D/DR)(R(DU/DR)) + (D/DZ)(DU/DZ) C C + LAMBDA*(1/R**2)*U = F(R,Z) C C THIS TWO-DIMENSIONAL MODIFIED HELMHOLTZ EQUATION RESULTS C FROM THE FOURIER TRANSFORM OF A THREE-DIMENSIONAL POISSON C EQUATION. 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 AND C A MUST BE NON-NEGATIVE. C C M C THE NUMBER OF GRID POINTS IN THE INTERVAL (A,B). THE GRID POINTS C IN THE R-DIRECTION ARE GIVEN BY R(I) = A + (I-0.5)DR FOR C I=1,2,...,M WHERE DR =(B-A)/M. M MUST BE GREATER THAN 2. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT R = A AND R = B. C C = 1 IF THE SOLUTION IS SPECIFIED AT R = A (SEE NOTE BELOW) AND C R = B. C C = 2 IF THE SOLUTION IS SPECIFIED AT R = A (SEE NOTE BELOW) AND C THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO R IS C SPECIFIED AT R = B. C 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 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 C = 5 IF THE SOLUTION IS UNSPECIFIED AT R = A = 0 AND THE SOLUTION C IS SPECIFIED AT R = B. C C = 6 IF THE SOLUTION IS UNSPECIFIED AT R = A = 0 AND THE C DERIVATIVE OF THE SOLUTION WITH RESPECT TO R IS SPECIFIED AT C R = B. C C NOTE: IF A = 0, DO NOT USE MBDCND = 1,2,3, OR 4, BUT INSTEAD C USE MBDCND = 5 OR 6. THE RESULTING APPROXIMATION GIVES C THE ONLY MEANINGFUL BOUNDARY CONDITION, I.E. DU/DR = 0. C (SEE D. GREENSPAN, 'INTRODUCTORY NUMERICAL ANALYSIS OF C ELLIPTIC BOUNDARY VALUE PROBLEMS,' HARPER AND ROW, 1965, C CHAPTER 5.) C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES (IF ANY) OF THE SOLUTION AT R = A. WHEN MBDCND = 1 OR 2, C C BDA(J) = U(A,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 3 OR 4, C C BDA(J) = (D/DR)U(A,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 5 OR 6, BDA IS A DUMMY VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT R = B. WHEN MBDCND = 1,4, OR 5, C C BDB(J) = U(B,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 2,3, OR 6, C C BDB(J) = (D/DR)U(B,Z(J)) , J=1,2,...,N. C C C,D C THE RANGE OF Z, I.E. C .LE. Z .LE. D. C MUST BE LESS C THAN D. C C N C THE NUMBER OF UNKNOWNS IN THE INTERVAL (C,D). THE UNKNOWNS IN C THE Z-DIRECTION ARE GIVEN BY Z(J) = C + (J-0.5)DZ, C J=1,2,...,N, WHERE DZ = (D-C)/N. N MUST BE GREATER THAN 2. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT Z = C C AND Z = D. C C = 0 IF THE SOLUTION IS PERIODIC IN Z, I.E. C U(I,J) = U(I,N+J). C C = 1 IF THE SOLUTION IS SPECIFIED AT Z = C AND Z = D. C C = 2 IF THE SOLUTION IS SPECIFIED AT Z = C AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO Z IS SPECIFIED AT C Z = D. C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Z IS C SPECIFIED AT Z = C AND Z = D. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Z IS C SPECIFIED AT Z = C AND THE SOLUTION IS SPECIFIED AT C Z = D. C C BDC C A ONE DIMENSIONAL ARRAY OF LENGTH M THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT Z = C. WHEN NBDCND = 1 OR 2, C C BDC(I) = U(R(I),C) , I=1,2,...,M. C C WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DZ)U(R(I),C), I=1,2,...,M. C C WHEN NBDCND = 0, BDC IS A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT Z = D. WHEN NBDCND = 1 OR 4, C C BDD(I) = U(R(I),D) , I=1,2,...,M. C C WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DZ)U(R(I),D) , I=1,2,...,M. C C WHEN NBDCND = 0, BDD IS A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE MODIFIED HELMHOLTZ EQUATION. IF C LAMBDA IS GREATER THAN 0, A SOLUTION MAY NOT EXIST. HOWEVER, C HSTCYL WILL ATTEMPT TO FIND A SOLUTION. LAMBDA MUST BE ZERO C WHEN MBDCND = 5 OR 6. C C F C A TWO-DIMENSIONAL ARRAY THAT SPECIFIES THE VALUES OF THE RIGHT C SIDE OF THE MODIFIED HELMHOLTZ EQUATION. FOR I=1,2,...,M C AND J=1,2,...,N C C F(I,J) = F(R(I),Z(J)) . C C F MUST BE DIMENSIONED AT LEAST M X N. C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY F AS IT APPEARS IN THE C PROGRAM CALLING HSTCYL. THIS PARAMETER IS USED TO SPECIFY THE C VARIABLE DIMENSION OF F. IDIMF MUST BE AT LEAST M. C C W C A ONE-DIMENSIONAL ARRAY THAT MUST BE PROVIDED BY THE USER FOR C WORK SPACE. W MAY REQUIRE UP TO 13M + 4N + M*INT(LOG2(N)) C LOCATIONS. THE ACTUAL NUMBER OF LOCATIONS USED IS COMPUTED BY C HSTCYL AND IS RETURNED IN THE LOCATION 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),Z(J)) FOR C I=1,2,...,M, J=1,2,...,N. 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 CON- C STANT, CALCULATED AND SUBTRACTED FROM F, WHICH ENSURES C THAT A SOLUTION EXISTS. HSTCYL THEN COMPUTES THIS C SOLUTION, WHICH IS A LEAST SQUARES SOLUTION TO THE C ORIGINAL APPROXIMATION. THIS SOLUTION PLUS ANY CONSTANT IS ALSO C A SOLUTION; HENCE, THE SOLUTION IS NOT UNIQUE. THE VALUE OF C PERTRB SHOULD BE SMALL COMPARED TO THE RIGHT SIDE F. C OTHERWISE, A SOLUTION IS OBTAINED TO AN ESSENTIALLY DIFFERENT C PROBLEM. THIS COMPARISON SHOULD ALWAYS BE MADE TO INSURE THAT C A MEANINGFUL SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. C EXCEPT TO NUMBERS 0 AND 11, A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR C C = 1 A .LT. 0 C C = 2 A .GE. B C C = 3 MBDCND .LT. 1 OR MBDCND .GT. 6 C C = 4 C .GE. D C C = 5 N .LE. 2 C C = 6 NBDCND .LT. 0 OR NBDCND .GT. 4 C C = 7 A = 0 AND MBDCND = 1,2,3, OR 4 C C = 8 A .GT. 0 AND MBDCND .GE. 5 C C = 9 M .LE. 2 C C = 10 IDIMF .LT. M C C = 11 LAMBDA .GT. 0 C C = 12 A=0, MBDCND .GE. 5, ELMBDA .NE. 0 C C SINCE THIS IS THE ONLY MEANS OF INDICATING A POSSIBLY C INCORRECT CALL TO HSTCYL, THE USER SHOULD TEST IERROR AFTER C 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),BDB(N),BDC(M),BDD(M),F(IDIMF,N), C ARGUMENTS W(SEE ARGUMENT LIST) C C LATEST JUNE 1, 1977 C REVISION C C SUBPROGRAMS HSTCYL,POISTG,POSTG2,GENBUN,POISD2,POISN2,POISP2, C REQUIRED COSGEN,MERGE,TRIX,TRI3,PIMACH C C SPECIAL NONE C CONDITIONS C C COMMON NONE C BLOCKS C C I/O NONE C C PRECISION SINGLE C C SPECIALIST ROLAND SWEET C C LANGUAGE FORTRAN C C HISTORY WRITTEN BY ROLAND SWEET AT NCAR IN MARCH, 1977 C C ALGORITHM THIS SUBROUTINE DEFINES THE FINITE-DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, ADJUSTS THE C RIGHT SIDE WHEN THE SYSTEM IS SINGULAR AND CALLS C EITHER POISTG OR GENBUN WHICH SOLVES THE LINEAR C SYSTEM OF EQUATIONS. C C SPACE 8228(DECIMAL) = 20044(OCTAL) LOCATIONS ON THE C REQUIRED NCAR CONTROL DATA 7600 C C TIMING AND THE EXECUTION TIME T ON THE NCAR CONTROL DATA C ACCURACY 7600 FOR SUBROUTINE HSTCYL IS ROUGHLY PROPORTIONAL C TO M*N*LOG2(N). SOME TYPICAL VALUES ARE LISTED IN C THE TABLE BELOW. C THE SOLUTION PROCESS EMPLOYED RESULTS IN A LOSS C OF NO MORE THAN FOUR SIGNIFICANT DIGITS FOR N AND M C AS LARGE AS 64. MORE DETAILED INFORMATION ABOUT C ACCURACY CAN BE FOUND IN THE DOCUMENTATION FOR C SUBROUTINE POISTG WHICH IS THE ROUTINE THAT C ACTUALLY SOLVES THE FINITE DIFFERENCE EQUATIONS. C C C M(=N) MBDCND NBDCND T(MSECS) C ----- ------ ------ -------- C C 32 1-6 1-4 56 C 64 1-6 1-4 230 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 SCHUMANN, U. AND R. SWEET,"A DIRECT METHOD FOR C THE SOLUTION OF POISSON"S EQUATION WITH NEUMANN C BOUNDARY CONDITIONS ON A STAGGERED GRID OF C ARBITRARY SIZE," J. COMP. PHYS. 20(1976), C PP. 171-182. C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C DIMENSION F(IDIMF,1) ,BDA(1) ,BDB(1) ,BDC(1) , 1 BDD(1) ,W(1) 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. 2) IERROR = 5 IF (NBDCND.LT.0 .OR. NBDCND.GE.5) IERROR = 6 IF (A.EQ.0. .AND. MBDCND.NE.5 .AND. MBDCND.NE.6) IERROR = 7 IF (A.GT.0. .AND. MBDCND.GE.5) IERROR = 8 IF (IDIMF .LT. M) IERROR = 10 IF (M .LE. 2) IERROR = 9 IF (A.EQ.0. .AND. MBDCND.GE.5 .AND. ELMBDA.NE.0.) IERROR = 12 IF (IERROR .NE. 0) RETURN DELTAR = (B-A)/FLOAT(M) DLRSQ = DELTAR**2 DELTHT = (D-C)/FLOAT(N) DLTHSQ = DELTHT**2 NP = NBDCND+1 C C DEFINE A,B,C COEFFICIENTS IN W-ARRAY. C IWB = M IWC = IWB+M IWR = IWC+M DO 101 I=1,M J = IWR+I W(J) = A+(FLOAT(I)-0.5)*DELTAR W(I) = (A+FLOAT(I-1)*DELTAR)/(DLRSQ*W(J)) K = IWC+I W(K) = (A+FLOAT(I)*DELTAR)/(DLRSQ*W(J)) K = IWB+I W(K) = ELMBDA/W(J)**2-2./DLRSQ 101 CONTINUE C C ENTER BOUNDARY DATA FOR R-BOUNDARIES. C GO TO (102,102,104,104,106,106),MBDCND 102 A1 = 2.*W(1) W(IWB+1) = W(IWB+1)-W(1) DO 103 J=1,N F(1,J) = F(1,J)-A1*BDA(J) 103 CONTINUE GO TO 106 104 A1 = DELTAR*W(1) W(IWB+1) = W(IWB+1)+W(1) DO 105 J=1,N F(1,J) = F(1,J)+A1*BDA(J) 105 CONTINUE 106 CONTINUE GO TO (107,109,109,107,107,109),MBDCND 107 W(IWC) = W(IWC)-W(IWR) A1 = 2.*W(IWR) DO 108 J=1,N F(M,J) = F(M,J)-A1*BDB(J) 108 CONTINUE GO TO 111 109 W(IWC) = W(IWC)+W(IWR) A1 = DELTAR*W(IWR) DO 110 J=1,N F(M,J) = F(M,J)-A1*BDB(J) 110 CONTINUE C C ENTER BOUNDARY DATA FOR THETA-BOUNDARIES. C 111 A1 = 2./DLTHSQ GO TO (121,112,112,114,114),NP 112 DO 113 I=1,M F(I,1) = F(I,1)-A1*BDC(I) 113 CONTINUE GO TO 116 114 A1 = 1./DELTHT DO 115 I=1,M F(I,1) = F(I,1)+A1*BDC(I) 115 CONTINUE 116 A1 = 2./DLTHSQ GO TO (121,117,119,119,117),NP 117 DO 118 I=1,M F(I,N) = F(I,N)-A1*BDD(I) 118 CONTINUE GO TO 121 119 A1 = 1./DELTHT DO 120 I=1,M F(I,N) = F(I,N)-A1*BDD(I) 120 CONTINUE 121 CONTINUE C C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A C SOLUTION. C PERTRB = 0. IF (ELMBDA) 130,123,122 122 IERROR = 11 GO TO 130 123 GO TO (130,130,124,130,130,124),MBDCND 124 GO TO (125,130,130,125,130),NP 125 CONTINUE DO 127 I=1,M A1 = 0. DO 126 J=1,N A1 = A1+F(I,J) 126 CONTINUE J = IWR+I PERTRB = PERTRB+A1*W(J) 127 CONTINUE PERTRB = PERTRB/(FLOAT(M*N)*0.5*(A+B)) DO 129 I=1,M DO 128 J=1,N F(I,J) = F(I,J)-PERTRB 128 CONTINUE 129 CONTINUE 130 CONTINUE C C MULTIPLY I-TH EQUATION THROUGH BY DELTHT**2 C DO 132 I=1,M W(I) = W(I)*DLTHSQ J = IWC+I W(J) = W(J)*DLTHSQ J = IWB+I W(J) = W(J)*DLTHSQ DO 131 J=1,N F(I,J) = F(I,J)*DLTHSQ 131 CONTINUE 132 CONTINUE LP = NBDCND W(1) = 0. W(IWR) = 0. C C CALL GENBUN TO SOLVE THE SYSTEM OF EQUATIONS. C IF (NBDCND .EQ. 0) GO TO 133 CALL POISTG (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1)) GO TO 134 133 CALL GENBUN (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1)) 134 CONTINUE W(1) = W(IWR+1)+3.*FLOAT(M) RETURN END