*DECK HWSPLR
SUBROUTINE HWSPLR (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND,
+ BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE HWSPLR
C***PURPOSE Solve a finite difference approximation to the Helmholtz
C equation in polar coordinates.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A1A
C***TYPE SINGLE PRECISION (HWSPLR-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 Subroutine HWSPLR solves a finite difference approximation to the
C Helmholtz equation in 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
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 condition 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,THETA(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,THETA(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 THETA, i.e., C .LE. THETA .LE. D. C must be less
C 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 THETA-direction given by THETA(J) = C+(J-1)DTHETA for
C J = 1,2,...,N+1, where DTHETA = (D-C)/N is the panel width. N
C must be greater than 3.
C
C NBDCND
C Indicates the type of boundary conditions at THETA = C and
C at THETA = D.
C
C = 0 If the solution is periodic in THETA, i.e.,
C U(I,J) = U(I,N+J).
C = 1 If the solution is specified at THETA = C and THETA = D
C (see note below).
C = 2 If the solution is specified at THETA = C and the
C derivative of the solution with respect to THETA is
C specified at THETA = D (see note below).
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
C (the former indicates that the solution is specified at
C R = 0, the latter indicates the solution is unspecified
C at R = 0). Use instead MBDCND = 1 or 2 .
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 THETA at
C THETA = C. When NBDCND = 3 or 4,
C
C BDC(I) = (d/dTHETA)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 THETA at
C THETA = D. When NBDCND = 2 or 3,
C
C BDD(I) = (d/dTHETA)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 .LT. 0, a solution may not exist. However, HWSPLR 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 and boundary values (if any).
C For I = 2,3,...,M and J = 2,3,...,N
C
C F(I,J) = F(R(I),THETA(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,THETA(J)) U(B,THETA(J))
C 2 U(A,THETA(J)) F(B,THETA(J))
C 3 F(A,THETA(J)) F(B,THETA(J))
C 4 F(A,THETA(J)) U(B,THETA(J)) J = 1,2,...,N+1
C 5 F(0,0) U(B,THETA(J))
C 6 F(0,0) F(B,THETA(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
C IDIMF
C The row (or first) dimension of the array F as it appears in the
C program calling HWSPLR. 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 HWSPLR 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),THETA(J)),
C I = 1,2,...,M+1, J = 1,2,...,N+1 .
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 constant,
C calculated and subtracted from F, which ensures that a solution
C exists. HWSPLR 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. PERTRB should be small compared to the right side.
C Otherwise, a solution is obtained to an essentially different
C problem. 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 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 .GT. 4 .
C = 7 A = 0, MBDCND = 3 or 4 .
C = 8 A .GT. 0, MBDCND .GE. 5 .
C = 9 MBDCND .GE. 5, NBDCND .NE. 0 and NBDCND .NE. 3 .
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 HWSPLR, 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 HWSPLR,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
C
C Precision Single
C
C Specialist Roland Sweet
C
C Language FORTRAN
C
C History Standardized April 1, 1973
C Revised January 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 13430(octal) = 5912(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 HWSPLR 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 HWSPLR
C
C
DIMENSION F(IDIMF,*)
DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) ,
1 W(*)
C***FIRST EXECUTABLE STATEMENT HWSPLR
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 (MBDCND.GE.5 .AND. NBDCND.NE.0 .AND. NBDCND.NE.3) 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 = MP1
GO TO (101,105,102,103,104,105),MBDCND
101 MSTOP = M
GO TO 105
102 MSTART = 1
GO TO 105
103 MSTART = 1
104 MSTOP = M
105 MUNK = MSTOP-MSTART+1
NSTART = 1
NSTOP = N
GO TO (109,106,107,108,109),NP
106 NSTART = 2
GO TO 109
107 NSTART = 2
108 NSTOP = NP1
109 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
A1 = 2./DLRSQ
IJ = 0
IF (MBDCND.EQ.3 .OR. MBDCND.EQ.4) IJ = 1
DO 110 I=1,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)
J = ID2+I
W(J) = -A1+ELMBDA
110 CONTINUE
GO TO (114,111,112,113,114,111),MBDCND
111 W(ID2) = A1
GO TO 114
112 W(ID2) = A1
113 W(ID3+1) = A1
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 THETA-BOUNDARIES.
C
124 A1 = 1./DLTHSQ
L = ID5-MSTART+1
LP = ID6-MSTART+1
GO TO (134,125,125,127,127),NP
125 DO 126 I=MSTART,MSTOP
J = I+LP
F(I,2) = F(I,2)-A1*W(J)*F(I,1)
126 CONTINUE
GO TO 129
127 A1 = 2./DELTHT
DO 128 I=MSTART,MSTOP
J = I+LP
F(I,1) = F(I,1)+A1*W(J)*BDC(I)
128 CONTINUE
129 A1 = 1./DLTHSQ
GO TO (134,130,132,132,130),NP
130 DO 131 I=MSTART,MSTOP
J = I+LP
F(I,N) = F(I,N)-A1*W(J)*F(I,NP1)
131 CONTINUE
GO TO 134
132 A1 = 2./DELTHT
DO 133 I=MSTART,MSTOP
J = I+LP
F(I,NP1) = F(I,NP1)-A1*W(J)*BDD(I)
133 CONTINUE
134 CONTINUE
C
C ADJUST RIGHT SIDE OF EQUATION FOR UNKNOWN AT POLE WHEN HAVE
C DERIVATIVE SPECIFIED BOUNDARY CONDITIONS.
C
IF (MBDCND.GE.5 .AND. NBDCND.EQ.3)
1 F(1,1) = F(1,1)-(BDD(2)-BDC(2))*4./(N*DELTHT*DLRSQ)
C
C ADJUST RIGHT SIDE OF SINGULAR PROBLEMS TO INSURE EXISTENCE OF A
C SOLUTION.
C
PERTRB = 0.
IF (ELMBDA) 144,136,135
135 IERROR = 11
GO TO 144
136 IF (NBDCND.NE.0 .AND. NBDCND.NE.3) GO TO 144
S2 = 0.
GO TO (144,144,137,144,144,138),MBDCND
137 W(ID5+1) = .5*(W(ID5+2)-DLRBY2)
S2 = .25*DELTAR
138 A2 = 2.
IF (NBDCND .EQ. 0) A2 = 1.
J = ID5+MUNK
W(J) = .5*(W(J-1)+DLRBY2)
S = 0.
DO 140 I=MSTART,MSTOP
S1 = 0.
IJ = NSTART+1
K = NSTOP-1
DO 139 J=IJ,K
S1 = S1+F(I,J)
139 CONTINUE
J = I+L
S = S+(A2*S1+F(I,NSTART)+F(I,NSTOP))*W(J)
140 CONTINUE
S2 = M*A+DELTAR*((M-1)*(M+1)*.5+.25)+S2
S1 = (2.+A2*(NUNK-2))*S2
IF (MBDCND .EQ. 3) GO TO 141
S2 = N*A2*DELTAR/8.
S = S+F(1,1)*S2
S1 = S1+S2
141 CONTINUE
PERTRB = S/S1
DO 143 I=MSTART,MSTOP
DO 142 J=NSTART,NSTOP
F(I,J) = F(I,J)-PERTRB
142 CONTINUE
143 CONTINUE
144 CONTINUE
C
C MULTIPLY I-TH EQUATION THROUGH BY (R(I)*DELTHT)**2.
C
DO 146 I=MSTART,MSTOP
K = I-MSTART+1
J = I+LP
A1 = DLTHSQ/W(J)
W(K) = A1*W(K)
J = ID2+K
W(J) = A1*W(J)
J = ID3+K
W(J) = A1*W(J)
DO 145 J=NSTART,NSTOP
F(I,J) = A1*F(I,J)
145 CONTINUE
146 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))
IWSTOR = W(ID4+1)+3*MUNK
GO TO (157,157,157,157,148,147),MBDCND
C
C ADJUST THE SOLUTION AS NECESSARY FOR THE PROBLEMS WHERE A = 0.
C
147 IF (ELMBDA .NE. 0.) GO TO 148
YPOLE = 0.
GO TO 155
148 CONTINUE
J = ID5+MUNK
W(J) = W(ID2)/W(ID3)
DO 149 IP=3,MUNK
I = MUNK-IP+2
J = ID5+I
LP = ID2+I
K = ID3+I
W(J) = W(I)/(W(LP)-W(K)*W(J+1))
149 CONTINUE
W(ID5+1) = -.5*DLTHSQ/(W(ID2+1)-W(ID3+1)*W(ID5+2))
DO 150 I=2,MUNK
J = ID5+I
W(J) = -W(J)*W(J-1)
150 CONTINUE
S = 0.
DO 151 J=NSTART,NSTOP
S = S+F(2,J)
151 CONTINUE
A2 = NUNK
IF (NBDCND .EQ. 0) GO TO 152
S = S-.5*(F(2,NSTART)+F(2,NSTOP))
A2 = A2-1.
152 YPOLE = (.25*DLRSQ*F(1,1)-S/A2)/(W(ID5+1)-1.+ELMBDA*DLRSQ*.25)
DO 154 I=MSTART,MSTOP
K = L+I
DO 153 J=NSTART,NSTOP
F(I,J) = F(I,J)+YPOLE*W(K)
153 CONTINUE
154 CONTINUE
155 DO 156 J=1,NP1
F(1,J) = YPOLE
156 CONTINUE
157 CONTINUE
IF (NBDCND .NE. 0) GO TO 159
DO 158 I=MSTART,MSTOP
F(I,NP1) = F(I,1)
158 CONTINUE
159 CONTINUE
W(1) = IWSTOR
RETURN
END