*DECK HWSCYL SUBROUTINE HWSCYL (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND, + BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W) C***BEGIN PROLOGUE HWSCYL C***PURPOSE Solve a standard finite difference approximation C to the Helmholtz equation in cylindrical coordinates. C***LIBRARY SLATEC (FISHPACK) C***CATEGORY I2B1A1A C***TYPE SINGLE PRECISION (HWSCYL-S) C***KEYWORDS CYLINDRICAL, ELLIPTIC, FISHPACK, HELMHOLTZ, PDE C***AUTHOR Adams, J., (NCAR) C Swarztrauber, P. N., (NCAR) C Sweet, R., (NCAR) C***DESCRIPTION C C Subroutine HWSCYL solves a finite difference approximation to the C Helmholtz equation in cylindrical coordinates: C C (1/R)(d/dR)(R(dU/dR)) + (d/dZ)(dU/dZ) C C + (LAMBDA/R**2)U = F(R,Z) C C This modified Helmholtz equation results from the Fourier C transform of the three-dimensional Poisson equation. 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 conditions 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,Z(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,Z(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 Z, i.e., C .LE. Z .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 Z-direction given by Z(J) = C+(J-1)DZ, for J = 1,2,...,N+1, C where DZ = (D-C)/N is the panel width. N must be greater than 3. C C NBDCND C Indicates the type of boundary conditions at Z = C and Z = D. C C = 0 If the solution is periodic in Z, i.e., U(I,1) = U(I,N+1). C = 1 If the solution is specified at Z = C and Z = D. C = 2 If the solution is specified at Z = C and the derivative of C the solution with respect to Z is specified at Z = D. C = 3 If the derivative of the solution with respect to Z is C specified at Z = C and Z = D. C = 4 If the derivative of the solution with respect to Z is C specified at Z = C and the solution is specified at Z = 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 Z at Z = C. C When NBDCND = 3 or 4, C C BDC(I) = (d/dZ)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 Z at Z = D. C When NBDCND = 2 or 3, C C BDD(I) = (d/dZ)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 .GT. 0, a solution may not exist. However, HWSCYL will C attempt to find a solution. LAMBDA must be zero when C MBDCND = 5 or 6 . C C F C A two-dimensional array that specifies the values of the right C side of the Helmholtz equation and boundary data (if any). For C I = 2,3,...,M and J = 2,3,...,N C C F(I,J) = F(R(I),Z(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,Z(J)) U(B,Z(J)) C 2 U(A,Z(J)) F(B,Z(J)) C 3 F(A,Z(J)) F(B,Z(J)) J = 1,2,...,N+1 C 4 F(A,Z(J)) U(B,Z(J)) C 5 F(0,Z(J)) U(B,Z(J)) C 6 F(0,Z(J)) F(B,Z(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 IDIMF C The row (or first) dimension of the array F as it appears in the C program calling HWSCYL. 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 HWSCYL 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),Z(J)), I = 1,2,...,M+1, C J = 1,2,...,N+1 . C C PERTRB C If one specifies a combination of periodic, derivative, and C unspecified boundary conditions 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. HWSCYL 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. The value of PERTRB should be small compared to the C right side F. Otherwise, a solution is obtained to an C essentially different problem. This comparison should always C be made to insure that a meaningful solution has been obtained. C C IERROR C An error flag which 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 NBDCND .GT. 4 . C = 7 A = 0, MBDCND = 3 or 4 . C = 8 A .GT. 0, MBDCND .GE. 5 . C = 9 A = 0, LAMBDA .NE. 0, MBDCND .GE. 5 . 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 HWSCYL, the user should test IERROR after the call. C C W C W(1) contains the required length of W. C C *Long Description: 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 HWSCYL,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 5818(decimal) = 13272(octal) 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 HWSCYL 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 The machine dependent constant PI is defined in 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 C***REFERENCES P. N. Swarztrauber and R. Sweet, Efficient Fortran C subprograms for the solution of elliptic equations, C NCAR TN/IA-109, July 1975, 138 pp. C***ROUTINES CALLED GENBUN C***REVISION HISTORY (YYMMDD) C 801001 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE HWSCYL C C DIMENSION F(IDIMF,*) DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) , 1 W(*) C***FIRST EXECUTABLE STATEMENT HWSCYL 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 (A.EQ.0. .AND. ELMBDA.NE.0. .AND. MBDCND.GE.5) 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)/M DLRBY2 = DELTAR/2. DLRSQ = DELTAR**2 NP1 = N+1 DELTHT = (D-C)/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 = M GO TO (104,103,102,101,101,102),MBDCND 101 MSTART = 1 GO TO 104 102 MSTART = 1 103 MSTOP = MP1 104 MUNK = MSTOP-MSTART+1 NSTART = 1 NSTOP = N GO TO (108,105,106,107,108),NP 105 NSTART = 2 GO TO 108 106 NSTART = 2 107 NSTOP = NP1 108 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 ISTART = 1 A1 = 2./DLRSQ IJ = 0 IF (MBDCND.EQ.3 .OR. MBDCND.EQ.4) IJ = 1 IF (MBDCND .LE. 4) GO TO 109 W(1) = 0. W(ID2+1) = -2.*A1 W(ID3+1) = 2.*A1 ISTART = 2 IJ = 1 109 DO 110 I=ISTART,MUNK R = A+(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) K = ID6+I J = ID2+I W(J) = -A1+ELMBDA*W(K) 110 CONTINUE GO TO (114,111,112,113,114,112),MBDCND 111 W(ID2) = A1 GO TO 114 112 W(ID2) = A1 113 W(ID3+1) = A1*ISTART 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 Z-BOUNDARIES. C 124 A1 = 1./DLTHSQ L = ID5-MSTART+1 GO TO (134,125,125,127,127),NP 125 DO 126 I=MSTART,MSTOP F(I,2) = F(I,2)-A1*F(I,1) 126 CONTINUE GO TO 129 127 A1 = 2./DELTHT DO 128 I=MSTART,MSTOP F(I,1) = F(I,1)+A1*BDC(I) 128 CONTINUE 129 A1 = 1./DLTHSQ GO TO (134,130,132,132,130),NP 130 DO 131 I=MSTART,MSTOP F(I,N) = F(I,N)-A1*F(I,NP1) 131 CONTINUE GO TO 134 132 A1 = 2./DELTHT DO 133 I=MSTART,MSTOP F(I,NP1) = F(I,NP1)-A1*BDD(I) 133 CONTINUE 134 CONTINUE C C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A C SOLUTION. C PERTRB = 0. IF (ELMBDA) 146,136,135 135 IERROR = 11 GO TO 146 136 W(ID5+1) = .5*(W(ID5+2)-DLRBY2) GO TO (146,146,138,146,146,137),MBDCND 137 W(ID5+1) = .5*W(ID5+1) 138 GO TO (140,146,146,139,146),NP 139 A2 = 2. GO TO 141 140 A2 = 1. 141 K = ID5+MUNK W(K) = .5*(W(K-1)+DLRBY2) S = 0. DO 143 I=MSTART,MSTOP S1 = 0. NSP1 = NSTART+1 NSTM1 = NSTOP-1 DO 142 J=NSP1,NSTM1 S1 = S1+F(I,J) 142 CONTINUE K = I+L S = S+(A2*S1+F(I,NSTART)+F(I,NSTOP))*W(K) 143 CONTINUE S2 = M*A+(.75+(M-1)*(M+1))*DLRBY2 IF (MBDCND .EQ. 3) S2 = S2+.25*DLRBY2 S1 = (2.+A2*(NUNK-2))*S2 PERTRB = S/S1 DO 145 I=MSTART,MSTOP DO 144 J=NSTART,NSTOP F(I,J) = F(I,J)-PERTRB 144 CONTINUE 145 CONTINUE 146 CONTINUE C C MULTIPLY I-TH EQUATION THROUGH BY DELTHT**2 TO PUT EQUATION INTO C CORRECT FORM FOR SUBROUTINE GENBUN. C DO 148 I=MSTART,MSTOP K = I-MSTART+1 W(K) = W(K)*DLTHSQ J = ID2+K W(J) = W(J)*DLTHSQ J = ID3+K W(J) = W(J)*DLTHSQ DO 147 J=NSTART,NSTOP F(I,J) = F(I,J)*DLTHSQ 147 CONTINUE 148 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)) W(1) = W(ID4+1)+3*MUNK IF (NBDCND .NE. 0) GO TO 150 DO 149 I=MSTART,MSTOP F(I,NP1) = F(I,1) 149 CONTINUE 150 CONTINUE RETURN END