*DECK HSTPLR
SUBROUTINE HSTPLR (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND,
+ BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE HSTPLR
C***PURPOSE Solve the standard five-point finite difference
C approximation on a staggered grid to the Helmholtz equation
C in polar coordinates.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A1A
C***TYPE SINGLE PRECISION (HSTPLR-S)
C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, POLAR
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
C
C HSTPLR solves the standard five-point finite difference
C approximation on a staggered grid to the Helmholtz equation in
C polar coordinates
C
C (1/R)(d/DR)(R(dU/DR)) + (1/R**2)(d/dTHETA)(dU/dTHETA)
C
C + LAMBDA*U = F(R,THETA)
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 and R = B.
C
C = 2 If the solution is specified at R = A and the derivative
C of the solution with respect to R is specified at R = B.
C (see note 1 below)
C
C = 3 If the derivative of the solution with respect to R is
C specified at R = A (see note 2 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 2 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 1: If A = 0, MBDCND = 2, and NBDCND = 0 or 3, the system of
C equations to be solved is singular. The unique solution
C is determined by extrapolation to the specification of
C U(0,THETA(1)). But in this case the right side of the
C system will be perturbed by the constant PERTRB.
C
C NOTE 2: 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 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,THETA(J)) , J=1,2,...,N.
C
C When MBDCND = 3 or 4,
C
C BDA(J) = (d/dR)U(A,THETA(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,THETA(J)) , J=1,2,...,N.
C
C When MBDCND = 2,3, or 6,
C
C BDB(J) = (d/dR)U(B,THETA(J)) , J=1,2,...,N.
C
C C,D
C The range of THETA, i.e. C .LE. THETA .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 THETA-direction are given by THETA(J) = C + (J-0.5)DT,
C J=1,2,...,N, where DT = (D-C)/N. N must be greater than 2.
C
C NBDCND
C Indicates the type of boundary conditions at THETA = C
C and THETA = D.
C
C = 0 If the solution is periodic in THETA, i.e.
C U(I,J) = U(I,N+J).
C
C = 1 If the solution is specified at THETA = C and THETA = D
C (see note below).
C
C = 2 If the solution is specified at THETA = C and the derivative
C of the solution with respect to THETA is specified at
C THETA = D (see note below).
C
C = 3 If the derivative of the solution with respect to THETA is
C specified at THETA = C and THETA = D.
C
C = 4 If the derivative of the solution with respect to THETA is
C specified at THETA = C and the solution is specified at
C THETA = d (see note below).
C
C NOTE: When NBDCND = 1, 2, or 4, do not use MBDCND = 5 or 6 (the
C former indicates that the solution is specified at R = 0; the
C latter indicates the solution is unspecified at R = 0). Use
C instead MBDCND = 1 or 2.
C
C BDC
C A one dimensional array of length M that specifies the boundary
C values of the solution at THETA = 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/dTHETA)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 THETA = 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/dTHETA)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 Helmholtz equation. If LAMBDA is
C greater than 0, a solution may not exist. However, HSTPLR 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(R(I),THETA(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 HSTPLR. 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 HSTPLR 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),THETA(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. HSTPLR 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 = 3 or 4
C
C = 8 A .GT. 0 and MBDCND .GE. 5
C
C = 9 MBDCND .GE. 5 and NBDCND .NE. 0 or 3
C
C = 10 IDIMF .LT. M
C
C = 11 LAMBDA .GT. 0
C
C = 12 M .LE. 2
C
C Since this is the only means of indicating a possibly
C incorrect call to HSTPLR, 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 HSTPLR,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 February, 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 8265(decimal) = 20111(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 HSTPLR 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 HSTPLR
C
C
DIMENSION F(IDIMF,*)
DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) ,
1 W(*)
C***FIRST EXECUTABLE STATEMENT HSTPLR
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.EQ.3 .OR. MBDCND.EQ.4)) IERROR = 7
IF (A.GT.0. .AND. MBDCND.GE.5) IERROR = 8
IF (MBDCND.GE.5 .AND. NBDCND.NE.0 .AND. NBDCND.NE.3) IERROR = 9
IF (IDIMF .LT. M) IERROR = 10
IF (M .LE. 2) IERROR = 12
IF (IERROR .NE. 0) RETURN
DELTAR = (B-A)/M
DLRSQ = DELTAR**2
DELTHT = (D-C)/N
DLTHSQ = DELTHT**2
NP = NBDCND+1
ISW = 1
MB = MBDCND
IF (A.EQ.0. .AND. MBDCND.EQ.2) MB = 6
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
K = IWC+I
W(K) = (A+I*DELTAR)/DLRSQ
K = IWB+I
W(K) = (ELMBDA-2./DLRSQ)*W(J)
101 CONTINUE
DO 103 I=1,M
J = IWR+I
A1 = W(J)
DO 102 J=1,N
F(I,J) = A1*F(I,J)
102 CONTINUE
103 CONTINUE
C
C ENTER BOUNDARY DATA FOR R-BOUNDARIES.
C
GO TO (104,104,106,106,108,108),MB
104 A1 = 2.*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
GO TO 108
106 A1 = DELTAR*W(1)
W(IWB+1) = W(IWB+1)+W(1)
DO 107 J=1,N
F(1,J) = F(1,J)+A1*BDA(J)
107 CONTINUE
108 GO TO (109,111,111,109,109,111),MB
109 A1 = 2.*W(IWR)
W(IWC) = W(IWC)-W(IWR)
DO 110 J=1,N
F(M,J) = F(M,J)-A1*BDB(J)
110 CONTINUE
GO TO 113
111 A1 = DELTAR*W(IWR)
W(IWC) = W(IWC)+W(IWR)
DO 112 J=1,N
F(M,J) = F(M,J)-A1*BDB(J)
112 CONTINUE
C
C ENTER BOUNDARY DATA FOR THETA-BOUNDARIES.
C
113 A1 = 2./DLTHSQ
GO TO (123,114,114,116,116),NP
114 DO 115 I=1,M
J = IWR+I
F(I,1) = F(I,1)-A1*BDC(I)/W(J)
115 CONTINUE
GO TO 118
116 A1 = 1./DELTHT
DO 117 I=1,M
J = IWR+I
F(I,1) = F(I,1)+A1*BDC(I)/W(J)
117 CONTINUE
118 A1 = 2./DLTHSQ
GO TO (123,119,121,121,119),NP
119 DO 120 I=1,M
J = IWR+I
F(I,N) = F(I,N)-A1*BDD(I)/W(J)
120 CONTINUE
GO TO 123
121 A1 = 1./DELTHT
DO 122 I=1,M
J = IWR+I
F(I,N) = F(I,N)-A1*BDD(I)/W(J)
122 CONTINUE
123 CONTINUE
C
C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A
C SOLUTION.
C
PERTRB = 0.
IF (ELMBDA) 133,125,124
124 IERROR = 11
GO TO 133
125 GO TO (133,133,126,133,133,126),MB
126 GO TO (127,133,133,127,133),NP
127 CONTINUE
ISW = 2
DO 129 J=1,N
DO 128 I=1,M
PERTRB = PERTRB+F(I,J)
128 CONTINUE
129 CONTINUE
PERTRB = PERTRB/(M*N*0.5*(A+B))
DO 131 I=1,M
J = IWR+I
A1 = PERTRB*W(J)
DO 130 J=1,N
F(I,J) = F(I,J)-A1
130 CONTINUE
131 CONTINUE
A2 = 0.
DO 132 J=1,N
A2 = A2+F(1,J)
132 CONTINUE
A2 = A2/W(IWR+1)
133 CONTINUE
C
C MULTIPLY I-TH EQUATION THROUGH BY R(I)*DELTHT**2
C
DO 135 I=1,M
J = IWR+I
A1 = DLTHSQ*W(J)
W(I) = A1*W(I)
J = IWC+I
W(J) = A1*W(J)
J = IWB+I
W(J) = A1*W(J)
DO 134 J=1,N
F(I,J) = A1*F(I,J)
134 CONTINUE
135 CONTINUE
LP = NBDCND
W(1) = 0.
W(IWR) = 0.
C
C CALL POISTG OR GENBUN TO SOLVE THE SYSTEM OF EQUATIONS.
C
IF (LP .EQ. 0) GO TO 136
CALL POISTG (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1))
GO TO 137
136 CALL GENBUN (LP,N,1,M,W,W(IWB+1),W(IWC+1),IDIMF,F,IERR1,W(IWR+1))
137 CONTINUE
W(1) = W(IWR+1)+3*M
IF (A.NE.0. .OR. MBDCND.NE.2 .OR. ISW.NE.2) GO TO 141
A1 = 0.
DO 138 J=1,N
A1 = A1+F(1,J)
138 CONTINUE
A1 = (A1-DLRSQ*A2/16.)/N
IF (NBDCND .EQ. 3) A1 = A1+(BDD(1)-BDC(1))/(D-C)
A1 = BDA(1)-A1
DO 140 I=1,M
DO 139 J=1,N
F(I,J) = F(I,J)+A1
139 CONTINUE
140 CONTINUE
141 CONTINUE
RETURN
END