C FISHPAK7 FROM PORTLIB 12/30/83 SUBROUTINE HSTCRT (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 HSTCRT SOLVES THE STANDARD FIVE-POINT FINITE DIFFERENCE C APPROXIMATION ON A STAGGERED GRID TO THE HELMHOLTZ EQUATION IN C CARTESIAN COORDINATES C C (D/DX)(DU/DX) + (D/DY)(DU/DY) + LAMBDA*U = F(X,Y) C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C * * * * * * * * PARAMETER DESCRIPTION * * * * * * * * * * C C * * * * * * ON INPUT * * * * * * C C A,B C THE RANGE OF X, I.E. A .LE. X .LE. B. A MUST BE LESS THAN B. C C M C THE NUMBER OF GRID POINTS IN THE INTERVAL (A,B). THE GRID POINTS C IN THE X-DIRECTION ARE GIVEN BY X(I) = A + (I-0.5)DX FOR C I=1,2,...,M WHERE DX =(B-A)/M. M MUST BE GREATER THAN 2. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT X = A AND X = B. C C = 0 IF THE SOLUTION IS PERIODIC IN X, C U(M+I,J) = U(I,J). C C = 1 IF THE SOLUTION IS SPECIFIED AT X = A AND X = B. C C = 2 IF THE SOLUTION IS SPECIFIED AT X = A AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO X IS SPECIFIED AT X = B. C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO X IS C SPECIFIED AT X = A AND X = B. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO X IS C SPECIFIED AT X = A AND THE SOLUTION IS SPECIFIED AT X = B. C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES (IF ANY) OF THE SOLUTION AT X = A. WHEN MBDCND = 1 OR 2, C C BDA(J) = U(A,Y(J)) , J=1,2,...,N. C C WHEN MBDCND = 3 OR 4, C C BDA(J) = (D/DX)U(A,Y(J)) , J=1,2,...,N. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT X = B. WHEN MBDCND = 1 OR 4 C C BDB(J) = U(B,Y(J)) , J=1,2,...,N. C C WHEN MBDCND = 2 OR 3 C C BDB(J) = (D/DX)U(B,Y(J)) , J=1,2,...,N. C C C,D C THE RANGE OF Y, I.E. C .LE. Y .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 Y-DIRECTION ARE GIVEN BY Y(J) = C + (J-0.5)DY, C J=1,2,...,N, WHERE DY = (D-C)/N. N MUST BE GREATER THAN 2. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT Y = C C AND Y = D. C C = 0 IF THE SOLUTION IS PERIODIC IN Y, I.E. C U(I,J) = U(I,N+J). C C = 1 IF THE SOLUTION IS SPECIFIED AT Y = C AND Y = D. C C = 2 IF THE SOLUTION IS SPECIFIED AT Y = C AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO Y IS SPECIFIED AT Y = D. C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Y IS C SPECIFIED AT Y = C AND Y = D. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Y IS C SPECIFIED AT Y = C AND THE SOLUTION IS SPECIFIED AT Y = D. C C BDC C A ONE DIMENSIONAL ARRAY OF LENGTH M THAT SPECIFIES THE BOUNDARY C VALUES OF THE SOLUTION AT Y = C. WHEN NBDCND = 1 OR 2, C C BDC(I) = U(X(I),C) , I=1,2,...,M. C C WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DY)U(X(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 Y = D. WHEN NBDCND = 1 OR 4, C C BDD(I) = U(X(I),D) , I=1,2,...,M. C C WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DY)U(X(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 HELMHOLTZ EQUATION. IF LAMBDA IS C GREATER THAN 0, A SOLUTION MAY NOT EXIST. HOWEVER, HSTCRT 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. FOR I=1,2,...,M AND J=1,2,...,N C C F(I,J) = F(X(I),Y(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 HSTCRT. 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 HSTCRT 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 (X(I),Y(J)) FOR C I=1,2,...,M, J=1,2,...,N. C C PERTRB C IF A COMBINATION OF PERIODIC OR DERIVATIVE BOUNDARY CONDITIONS IS C SPECIFIED FOR A POISSON EQUATION (LAMBDA = 0), A SOLUTION MAY NOT C EXIST. PERTRB IS A CONSTANT, CALCULATED AND SUBTRACTED FROM F, C WHICH ENSURES THAT A SOLUTION EXISTS. HSTCRT THEN COMPUTES THIS C SOLUTION, WHICH IS A LEAST SQUARES SOLUTION TO THE ORIGINAL C APPROXIMATION. THIS SOLUTION PLUS ANY CONSTANT IS ALSO A C SOLUTION; HENCE, THE SOLUTION IS NOT UNIQUE. THE VALUE OF PERTRB C SHOULD BE SMALL COMPARED TO THE RIGHT SIDE F. OTHERWISE, A C SOLUTION IS OBTAINED TO AN ESSENTIALLY DIFFERENT PROBLEM. THIS C COMPARISON SHOULD ALWAYS BE MADE TO INSURE THAT A MEANINGFUL C SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. C EXCEPT TO NUMBERS 0 AND 6, A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR C C = 1 A .GE. B C C = 2 MBDCND .LT. 0 OR MBDCND .GT. 4 C C = 3 C .GE. D C C = 4 N .LE. 2 C C = 5 NBDCND .LT. 0 OR NBDCND .GT. 4 C C = 6 LAMBDA .GT. 0 C C = 7 IDIMF .LT. M C C = 8 M .LE. 2 C C SINCE THIS IS THE ONLY MEANS OF INDICATING A POSSIBLY C INCORRECT CALL TO HSTCRT, 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 HSTCRT,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 JANUARY , 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 8131(DECIMAL) = 17703(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 HSTCRT 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 1-4 1-4 56 C 64 1-4 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) 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. 2) IERROR = 4 IF (NBDCND.LT.0 .OR. NBDCND.GT.4) IERROR = 5 IF (IDIMF .LT. M) IERROR = 7 IF (M .LE. 2) IERROR = 8 IF (IERROR .NE. 0) RETURN NPEROD = NBDCND MPEROD = 0 IF (MBDCND .GT. 0) MPEROD = 1 DELTAX = (B-A)/FLOAT(M) TWDELX = 1./DELTAX DELXSQ = 2./DELTAX**2 DELTAY = (D-C)/FLOAT(N) TWDELY = 1./DELTAY DELYSQ = DELTAY**2 TWDYSQ = 2./DELYSQ NP = NBDCND+1 MP = MBDCND+1 C C DEFINE THE A,B,C COEFFICIENTS IN W-ARRAY. C ID2 = M ID3 = ID2+M ID4 = ID3+M S = (DELTAY/DELTAX)**2 ST2 = 2.*S DO 101 I=1,M W(I) = S J = ID2+I W(J) = -ST2+ELMBDA*DELYSQ J = ID3+I W(J) = S 101 CONTINUE C C ENTER BOUNDARY DATA FOR X-BOUNDARIES. C GO TO (111,102,102,104,104),MP 102 DO 103 J=1,N F(1,J) = F(1,J)-BDA(J)*DELXSQ 103 CONTINUE W(ID2+1) = W(ID2+1)-W(1) GO TO 106 104 DO 105 J=1,N F(1,J) = F(1,J)+BDA(J)*TWDELX 105 CONTINUE W(ID2+1) = W(ID2+1)+W(1) 106 GO TO (111,107,109,109,107),MP 107 DO 108 J=1,N F(M,J) = F(M,J)-BDB(J)*DELXSQ 108 CONTINUE W(ID3) = W(ID3)-W(1) GO TO 111 109 DO 110 J=1,N F(M,J) = F(M,J)-BDB(J)*TWDELX 110 CONTINUE W(ID3) = W(ID3)+W(1) 111 CONTINUE C C ENTER BOUNDARY DATA FOR Y-BOUNDARIES. C GO TO (121,112,112,114,114),NP 112 DO 113 I=1,M F(I,1) = F(I,1)-BDC(I)*TWDYSQ 113 CONTINUE GO TO 116 114 DO 115 I=1,M F(I,1) = F(I,1)+BDC(I)*TWDELY 115 CONTINUE 116 GO TO (121,117,119,119,117),NP 117 DO 118 I=1,M F(I,N) = F(I,N)-BDD(I)*TWDYSQ 118 CONTINUE GO TO 121 119 DO 120 I=1,M F(I,N) = F(I,N)-BDD(I)*TWDELY 120 CONTINUE 121 CONTINUE DO 123 I=1,M DO 122 J=1,N F(I,J) = F(I,J)*DELYSQ 122 CONTINUE 123 CONTINUE IF (MPEROD .EQ. 0) GO TO 124 W(1) = 0. W(ID4) = 0. 124 CONTINUE PERTRB = 0. IF (ELMBDA) 133,126,125 125 IERROR = 6 GO TO 133 126 GO TO (127,133,133,127,133),MP 127 GO TO (128,133,133,128,133),NP C C FOR SINGULAR PROBLEMS MUST ADJUST DATA TO INSURE THAT A SOLUTION C WILL EXIST. C 128 CONTINUE S = 0. DO 130 J=1,N DO 129 I=1,M S = S+F(I,J) 129 CONTINUE 130 CONTINUE PERTRB = S/FLOAT(M*N) DO 132 J=1,N DO 131 I=1,M F(I,J) = F(I,J)-PERTRB 131 CONTINUE 132 CONTINUE PERTRB = PERTRB/DELYSQ C C SOLVE THE EQUATION. C 133 CONTINUE IF (NPEROD .EQ. 0) GO TO 134 CALL POISTG (NPEROD,N,MPEROD,M,W(1),W(ID2+1),W(ID3+1),IDIMF,F, 1 IERR1,W(ID4+1)) GO TO 135 134 CONTINUE CALL GENBUN (NPEROD,N,MPEROD,M,W(1),W(ID2+1),W(ID3+1),IDIMF,F, 1 IERR1,W(ID4+1)) 135 CONTINUE W(1) = W(ID4+1)+3.*FLOAT(M) RETURN END