*DECK HSTCYL
SUBROUTINE HSTCYL (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND,
+ BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE HSTCYL
C***PURPOSE Solve the standard five-point finite difference
C approximation on a staggered grid to the modified
C Helmholtz equation in cylindrical coordinates.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A1A
C***TYPE SINGLE PRECISION (HSTCYL-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 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 for 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 *Long Description:
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 The machine dependent constant PI is defined in
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
C***REFERENCES U. Schumann and R. Sweet, A direct method for the
C solution of Poisson's equation with Neumann boundary
C conditions on a staggered grid of arbitrary size,
C Journal of Computational Physics 20, (1976),
C pp. 171-182.
C***ROUTINES CALLED GENBUN, POISTG
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 HSTCYL
C
C
DIMENSION F(IDIMF,*) ,BDA(*) ,BDB(*) ,BDC(*) ,
1 BDD(*) ,W(*)
C***FIRST EXECUTABLE STATEMENT HSTCYL
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)/M
DLRSQ = DELTAR**2
DELTHT = (D-C)/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+(I-0.5)*DELTAR
W(I) = (A+(I-1)*DELTAR)/(DLRSQ*W(J))
K = IWC+I
W(K) = (A+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/(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*M
RETURN
END