*DECK HWSCRT
SUBROUTINE HWSCRT (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND,
+ BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE HWSCRT
C***PURPOSE Solves the standard five-point finite difference
C approximation to the Helmholtz equation in Cartesian
C coordinates.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A1A
C***TYPE SINGLE PRECISION (HWSCRT-S)
C***KEYWORDS CARTESIAN, ELLIPTIC, FISHPACK, HELMHOLTZ, PDE
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
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 *Long Description:
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 The machine dependent constant PI is defined in
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***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 HWSCRT
C
C
DIMENSION F(IDIMF,*)
DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) ,
1 W(*)
C***FIRST EXECUTABLE STATEMENT HWSCRT
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)/M
TWDELX = 2./DELTAX
DELXSQ = 1./DELTAX**2
DELTAY = (D-C)/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.+(NUNK-2)*A2)*(2.+(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*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