C FISHPAK2 FROM PORTLIB 12/30/83 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 * * 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 SUBROUTINE HWSCRT SOLVES THE STANDARD FIVE-POINT FINITE C DIFFERENCE APPROXIMATION TO THE HELMHOLTZ EQUATION IN CARTESIAN C 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 PANELS INTO WHICH THE INTERVAL (A,B) IS C SUBDIVIDED. HENCE, THERE WILL BE M+1 GRID POINTS IN THE C X-DIRECTION GIVEN BY X(I) = A+(I-1)DX FOR I = 1,2,...,M+1, C WHERE DX = (B-A)/M IS THE PANEL WIDTH. M MUST BE GREATER THAN 3. 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, I.E., U(I,J) = U(M+I,J). C = 1 IF THE SOLUTION IS SPECIFIED AT X = A AND X = B. C = 2 IF THE SOLUTION IS SPECIFIED AT X = A AND THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO X IS SPECIFIED AT X = B. C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO X IS C SPECIFIED AT X = A AND X = B. 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+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO X AT X = A. 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 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 X AT X = B. 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 DUMMY VARIABLE. C C C,D C THE RANGE OF Y, I.E., C .LE. Y .LE. D. C MUST BE LESS 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 Y-DIRECTION GIVEN BY Y(J) = C+(J-1)DY FOR J = 1,2,...,N+1, WHERE C DY = (D-C)/N IS THE PANEL WIDTH. N MUST BE GREATER THAN 3. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS AT Y = C AND Y = D. C C = 0 IF THE SOLUTION IS PERIODIC IN Y, I.E., U(I,J) = U(I,N+J). C = 1 IF THE SOLUTION IS SPECIFIED AT Y = C AND Y = D. C = 2 IF THE SOLUTION IS SPECIFIED AT Y = C AND THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO Y IS SPECIFIED AT Y = D. C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Y IS C SPECIFIED AT Y = C AND Y = D. 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+1 THAT SPECIFIES THE VALUES C OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Y AT Y = 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 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 Y AT Y = D. 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 A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE HELMHOLTZ EQUATION. IF C LAMBDA .GT. 0, A SOLUTION MAY NOT EXIST. HOWEVER, HWSCRT WILL C ATTEMPT TO FIND A SOLUTION. C C F C A TWO-DIMENSIONAL ARRAY WHICH 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(X(I),Y(J)). C C ON THE BOUNDARIES F IS DEFINED BY 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)) J = 1,2,...,N+1 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) I = 1,2,...,M+1 C 3 F(X(I),C) F(X(I),D) C 4 F(X(I),C) U(X(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 IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY F AS IT APPEARS IN THE C PROGRAM CALLING HWSCRT. 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 HWSCRT 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 (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 BOUNDARY CONDITIONS C IS SPECIFIED FOR A POISSON EQUATION (LAMBDA = 0), A SOLUTION MAY C NOT EXIST. PERTRB IS A CONSTANT, CALCULATED AND SUBTRACTED FROM C F, WHICH ENSURES THAT A SOLUTION EXISTS. HWSCRT THEN COMPUTES C THIS 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 C PERTRB SHOULD BE SMALL COMPARED TO THE RIGHT SIDE F. OTHERWISE, C A SOLUTION IS OBTAINED TO AN ESSENTIALLY DIFFERENT PROBLEM. C THIS COMPARISON SHOULD ALWAYS BE MADE TO INSURE THAT A C MEANINGFUL SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT PARAMETERS. EXCEPT C FOR NUMBERS 0 AND 6, 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 A POSSIBLY INCORRECT C CALL TO HWSCRT, 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 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 HWSCRT,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 NONE C C PRECISION SINGLE C C SPECIALIST ROLAND SWEET C C LANGUAGE FORTRAN C C HISTORY STANDARDIZED SEPTEMBER 1, 1973 C REVISED APRIL 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 13110(OCTAL) = 5704(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 HWSCRT 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 0 0 31 C 32 1 1 23 C 32 3 3 36 C 64 0 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 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 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 .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 END