*DECK SEPELI
SUBROUTINE SEPELI (INTL, IORDER, A, B, M, MBDCND, BDA, ALPHA, BDB,
+ BETA, C, D, N, NBDCND, BDC, GAMA, BDD, XNU, COFX, COFY, GRHS,
+ USOL, IDMN, W, PERTRB, IERROR)
C***BEGIN PROLOGUE SEPELI
C***PURPOSE Discretize and solve a second and, optionally, a fourth
C order finite difference approximation on a uniform grid to
C the general separable elliptic partial differential
C equation on a rectangle with any combination of periodic or
C mixed boundary conditions.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A2
C***TYPE SINGLE PRECISION (SEPELI-S)
C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, SEPARABLE
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
C
C Dimension of BDA(N+1), BDB(N+1), BDC(M+1), BDD(M+1),
C Arguments USOL(IDMN,N+1), GRHS(IDMN,N+1),
C W (see argument list)
C
C Latest Revision March 1977
C
C Purpose SEPELI solves for either the second-order
C finite difference approximation or a
C fourth-order approximation to a separable
C elliptic equation.
C
C 2 2
C AF(X)*d U/dX + BF(X)*dU/dX + CF(X)*U +
C 2 2
C DF(Y)*d U/dY + EF(Y)*dU/dY + FF(Y)*U
C
C = G(X,Y)
C
C on a rectangle (X greater than or equal to A
C and less than or equal to B; Y greater than
C or equal to C and less than or equal to D).
C Any combination of periodic or mixed boundary
C conditions is allowed.
C
C Purpose The possible boundary conditions are:
C in the X-direction:
C (0) Periodic, U(X+B-A,Y)=U(X,Y) for all Y,X
C (1) U(A,Y), U(B,Y) are specified for all Y
C (2) U(A,Y), dU(B,Y)/dX+BETA*U(B,Y) are
C specified for all Y
C (3) dU(A,Y)/dX+ALPHA*U(A,Y),dU(B,Y)/dX+
C BETA*U(B,Y) are specified for all Y
C (4) dU(A,Y)/dX+ALPHA*U(A,Y),U(B,Y) are
C specified for all Y
C
C in the Y-direction:
C (0) Periodic, U(X,Y+D-C)=U(X,Y) for all X,Y
C (1) U(X,C),U(X,D) are specified for all X
C (2) U(X,C),dU(X,D)/dY+XNU*U(X,D) are specified
C for all X
C (3) dU(X,C)/dY+GAMA*U(X,C),dU(X,D)/dY+
C XNU*U(X,D) are specified for all X
C (4) dU(X,C)/dY+GAMA*U(X,C),U(X,D) are
C specified for all X
C
C Arguments
C
C On Input INTL
C = 0 On initial entry to SEPELI or if any of
C the arguments C, D, N, NBDCND, COFY are
C changed from a previous call
C = 1 If C, D, N, NBDCND, COFY are unchanged
C from the previous call.
C
C IORDER
C = 2 If a second-order approximation is sought
C = 4 If a fourth-order approximation is sought
C
C A,B
C The range of the X-independent variable;
C i.e., X is greater than or equal to A and
C less than or equal to B. A must be less than
C B.
C
C M
C The number of panels into which the interval
C [A,B] is subdivided. Hence, there will be
C M+1 grid points in the X-direction given by
C XI=A+(I-1)*DLX for I=1,2,...,M+1 where
C DLX=(B-A)/M is the panel width. M must be
C less than IDMN and greater than 5.
C
C MBDCND
C Indicates the type of boundary condition at
C X=A and X=B
C = 0 If the solution is periodic in X; i.e.,
C U(X+B-A,Y)=U(X,Y) for all Y,X
C = 1 If the solution is specified at X=A and
C X=B; i.e., U(A,Y) and U(B,Y) are
C specified for all Y
C = 2 If the solution is specified at X=A and
C the boundary condition is mixed at X=B;
C i.e., U(A,Y) and dU(B,Y)/dX+BETA*U(B,Y)
C are specified for all Y
C = 3 If the boundary conditions at X=A and X=B
C are mixed; i.e., dU(A,Y)/dX+ALPHA*U(A,Y)
C and dU(B,Y)/dX+BETA*U(B,Y) are specified
C for all Y
C = 4 If the boundary condition at X=A is mixed
C and the solution is specified at X=B;
C i.e., dU(A,Y)/dX+ALPHA*U(A,Y) and U(B,Y)
C are specified for all Y
C
C BDA
C A one-dimensional array of length N+1 that
C specifies the values of dU(A,Y)/dX+
C ALPHA*U(A,Y) at X=A, when MBDCND=3 or 4.
C BDA(J) = dU(A,YJ)/dX+ALPHA*U(A,YJ);
C J=1,2,...,N+1
C when MBDCND has any other value, BDA is a
C dummy parameter.
C
C On Input ALPHA
C The scalar multiplying the solution in case
C of a mixed boundary condition at X=A (see
C argument BDA). If MBDCND = 3,4 then ALPHA is
C a dummy parameter.
C
C BDB
C A one-dimensional array of length N+1 that
C specifies the values of dU(B,Y)/dX+
C BETA*U(B,Y) at X=B. When MBDCND=2 or 3
C BDB(J) = dU(B,YJ)/dX+BETA*U(B,YJ);
C J=1,2,...,N+1
C When MBDCND has any other value, BDB is a
C dummy parameter.
C
C BETA
C The scalar multiplying the solution in case
C of a mixed boundary condition at X=B (see
C argument BDB). If MBDCND=2,3 then BETA is a
C dummy parameter.
C
C C,D
C The range of the Y-independent variable;
C i.e., Y is greater than or equal to C and
C less than or equal to D. C must be less than
C D.
C
C N
C The number of panels into which the interval
C [C,D] is subdivided. Hence, there will be
C N+1 grid points in the Y-direction given by
C YJ=C+(J-1)*DLY for J=1,2,...,N+1 where
C DLY=(D-C)/N is the panel width. In addition,
C N must be greater than 4.
C
C NBDCND
C Indicates the types of boundary conditions at
C Y=C and Y=D
C = 0 If the solution is periodic in Y; i.e.,
C U(X,Y+D-C)=U(X,Y) for all X,Y
C = 1 If the solution is specified at Y=C and
C Y = D, i.e., U(X,C) and U(X,D) are
C specified for all X
C = 2 If the solution is specified at Y=C and
C the boundary condition is mixed at Y=D;
C i.e., U(X,C) and dU(X,D)/dY+XNU*U(X,D)
C are specified for all X
C = 3 If the boundary conditions are mixed at
C Y=C and Y=D; i.e., dU(X,D)/dY+GAMA*U(X,C)
C and dU(X,D)/dY+XNU*U(X,D) are specified
C for all X
C = 4 If the boundary condition is mixed at Y=C
C and the solution is specified at Y=D;
C i.e. dU(X,C)/dY+GAMA*U(X,C) and U(X,D)
C are specified for all X
C
C BDC
C A one-dimensional array of length M+1 that
C specifies the value of dU(X,C)/dY+GAMA*U(X,C)
C at Y=C. When NBDCND=3 or 4
C BDC(I) = dU(XI,C)/dY + GAMA*U(XI,C);
C I=1,2,...,M+1.
C When NBDCND has any other value, BDC is a
C dummy parameter.
C
C GAMA
C The scalar multiplying the solution in case
C of a mixed boundary condition at Y=C (see
C argument BDC). If NBDCND=3,4 then GAMA is a
C dummy parameter.
C
C BDD
C A one-dimensional array of length M+1 that
C specifies the value of dU(X,D)/dY +
C XNU*U(X,D) at Y=C. When NBDCND=2 or 3
C BDD(I) = dU(XI,D)/dY + XNU*U(XI,D);
C I=1,2,...,M+1.
C When NBDCND has any other value, BDD is a
C dummy parameter.
C
C XNU
C The scalar multiplying the solution in case
C of a mixed boundary condition at Y=D (see
C argument BDD). If NBDCND=2 or 3 then XNU is
C a dummy parameter.
C
C COFX
C A user-supplied subprogram with
C parameters X, AFUN, BFUN, CFUN which
C returns the values of the X-dependent
C coefficients AF(X), BF(X), CF(X) in
C the elliptic equation at X.
C
C COFY
C A user-supplied subprogram with
C parameters Y, DFUN, EFUN, FFUN which
C returns the values of the Y-dependent
C coefficients DF(Y), EF(Y), FF(Y) in
C the elliptic equation at Y.
C
C NOTE: COFX and COFY must be declared external
C in the calling routine. The values returned in
C AFUN and DFUN must satisfy AFUN*DFUN greater
C than 0 for A less than X less than B,
C C less than Y less than D (see IERROR=10).
C The coefficients provided may lead to a matrix
C equation which is not diagonally dominant in
C which case solution may fail (see IERROR=4).
C
C GRHS
C A two-dimensional array that specifies the
C values of the right-hand side of the elliptic
C equation; i.e., GRHS(I,J)=G(XI,YI), for
C I=2,...,M; J=2,...,N. At the boundaries,
C GRHS is defined by
C
C MBDCND GRHS(1,J) GRHS(M+1,J)
C ------ --------- -----------
C 0 G(A,YJ) G(B,YJ)
C 1 * *
C 2 * G(B,YJ) J=1,2,...,N+1
C 3 G(A,YJ) G(B,YJ)
C 4 G(A,YJ) *
C
C NBDCND GRHS(I,1) GRHS(I,N+1)
C ------ --------- -----------
C 0 G(XI,C) G(XI,D)
C 1 * *
C 2 * G(XI,D) I=1,2,...,M+1
C 3 G(XI,C) G(XI,D)
C 4 G(XI,C) *
C
C where * means these quantities are not used.
C GRHS should be dimensioned IDMN by at least
C N+1 in the calling routine.
C
C USOL
C A two-dimensional array that specifies the
C values of the solution along the boundaries.
C At the boundaries, USOL is defined by
C
C MBDCND USOL(1,J) USOL(M+1,J)
C ------ --------- -----------
C 0 * *
C 1 U(A,YJ) U(B,YJ)
C 2 U(A,YJ) * J=1,2,...,N+1
C 3 * *
C 4 * U(B,YJ)
C
C NBDCND USOL(I,1) USOL(I,N+1)
C ------ --------- -----------
C 0 * *
C 1 U(XI,C) U(XI,D)
C 2 U(XI,C) * I=1,2,...,M+1
C 3 * *
C 4 * U(XI,D)
C
C where * means the quantities are not used in
C the solution.
C
C If IORDER=2, the user may equivalence GRHS
C and USOL to save space. Note that in this
C case the tables specifying the boundaries of
C the GRHS and USOL arrays determine the
C boundaries uniquely except at the corners.
C If the tables call for both G(X,Y) and
C U(X,Y) at a corner then the solution must be
C chosen. For example, if MBDCND=2 and
C NBDCND=4, then U(A,C), U(A,D), U(B,D) must be
C chosen at the corners in addition to G(B,C).
C
C If IORDER=4, then the two arrays, USOL and
C GRHS, must be distinct.
C
C USOL should be dimensioned IDMN by at least
C N+1 in the calling routine.
C
C IDMN
C The row (or first) dimension of the arrays
C GRHS and USOL as it appears in the program
C calling SEPELI. This parameter is used to
C specify the variable dimension of GRHS and
C USOL. IDMN must be at least 7 and greater
C than or equal to M+1.
C
C W
C A one-dimensional array that must be provided
C by the user for work space. Let
C K=INT(log2(N+1))+1 and set L=2**(K+1).
C then (K-2)*L+K+10*N+12*M+27 will suffice
C as a length of W. THE actual length of W in
C the calling routine must be set in W(1) (see
C IERROR=11).
C
C On Output USOL
C Contains the approximate solution to the
C elliptic equation. USOL(I,J) is the
C approximation to U(XI,YJ) for I=1,2...,M+1
C and J=1,2,...,N+1. The approximation has
C error O(DLX**2+DLY**2) if called with
C IORDER=2 and O(DLX**4+DLY**4) if called with
C IORDER=4.
C
C W
C Contains intermediate values that must not be
C destroyed if SEPELI is called again with
C INTL=1. In addition W(1) contains the exact
C minimal length (in floating point) required
C for the work space (see IERROR=11).
C
C PERTRB
C If a combination of periodic or derivative
C boundary conditions (i.e., ALPHA=BETA=0 if
C MBDCND=3; GAMA=XNU=0 if NBDCND=3) is
C specified and if the coefficients of U(X,Y)
C in the separable elliptic equation are zero
C (i.e., CF(X)=0 for X greater than or equal to
C A and less than or equal to B; FF(Y)=0 for
C Y greater than or equal to C and less than
C or equal to D) then a solution may not exist.
C PERTRB is a constant calculated and
C subtracted from the right-hand side of the
C matrix equations generated by SEPELI which
C insures that a solution exists. SEPELI then
C computes this solution which is a weighted
C minimal least squares solution to the
C original problem.
C
C IERROR
C An error flag that indicates invalid input
C parameters or failure to find a solution
C = 0 No error
C = 1 If A greater than B or C greater than D
C = 2 If MBDCND less than 0 or MBDCND greater
C than 4
C = 3 If NBDCND less than 0 or NBDCND greater
C than 4
C = 4 If attempt to find a solution fails.
C (the linear system generated is not
C diagonally dominant.)
C = 5 If IDMN is too small (see discussion of
C IDMN)
C = 6 If M is too small or too large (see
C discussion of M)
C = 7 If N is too small (see discussion of N)
C = 8 If IORDER is not 2 or 4
C = 9 If INTL is not 0 or 1
C = 10 If AFUN*DFUN less than or equal to 0 for
C some interior mesh point (XI,YJ)
C = 11 If the work space length input in W(1)
C is less than the exact minimal work
C space length required output in W(1).
C
C NOTE (concerning IERROR=4): for the
C coefficients input through COFX, COFY, the
C discretization may lead to a block
C tridiagonal linear system which is not
C diagonally dominant (for example, this
C happens if CFUN=0 and BFUN/(2.*DLX) greater
C than AFUN/DLX**2). In this case solution may
C fail. This cannot happen in the limit as
C DLX, DLY approach zero. Hence, the condition
C may be remedied by taking larger values for M
C or N.
C
C Entry Points SEPELI, SPELIP, CHKPRM, CHKSNG, ORTHOG, MINSOL,
C TRISP, DEFER, DX, DY, BLKTRI, BLKTR1, INDXB,
C INDXA, INDXC, PROD, PRODP, CPROD, CPRODP,
C PPADD, PSGF, BSRH, PPSGF, PPSPF, COMPB,
C TRUN1, STOR1, TQLRAT
C
C Special Conditions NONE
C
C Common Blocks SPLP, CBLKT
C
C I/O NONE
C
C Precision Single
C
C Specialist John C. Adams, NCAR, Boulder, Colorado 80307
C
C Language FORTRAN
C
C History Developed at NCAR during 1975-76.
C
C Algorithm SEPELI automatically discretizes the separable
C elliptic equation which is then solved by a
C generalized cyclic reduction algorithm in the
C subroutine, BLKTRI. The fourth-order solution
C is obtained using 'Deferred Corrections' which
C is described and referenced in sections,
C references and method.
C
C Space Required 14654 (octal) = 6572 (decimal)
C
C Accuracy and Timing The following computational results were
C obtained by solving the sample problem at the
C end of this write-up on the Control Data 7600.
C The op count is proportional to M*N*log2(N).
C In contrast to the other routines in this
C chapter, accuracy is tested by computing and
C tabulating second- and fourth-order
C discretization errors. Below is a table
C containing computational results. The times
C given do not include initialization (i.e.,
C times are for INTL=1). Note that the
C fourth-order accuracy is not realized until the
C mesh is sufficiently refined.
C
C Second-order Fourth-order Second-order Fourth-order
C M N Execution Time Execution Time Error Error
C (M SEC) (M SEC)
C 6 6 6 14 6.8E-1 1.2E0
C 14 14 23 58 1.4E-1 1.8E-1
C 30 30 100 247 3.2E-2 9.7E-3
C 62 62 445 1,091 7.5E-3 3.0E-4
C 126 126 2,002 4,772 1.8E-3 3.5E-6
C
C Portability There are no machine-dependent constants.
C
C Required Resident SQRT, ABS, LOG
C Routines
C
C References Keller, H.B., 'Numerical Methods for Two-point
C Boundary-value Problems', Blaisdel (1968),
C Waltham, Mass.
C
C Swarztrauber, P., and R. Sweet (1975):
C 'Efficient FORTRAN Subprograms for The
C Solution of Elliptic Partial Differential
C Equations'. NCAR Technical Note
C NCAR-TN/IA-109, pp. 135-137.
C
C***REFERENCES H. B. Keller, Numerical Methods for Two-point
C Boundary-value Problems, Blaisdel, Waltham, Mass.,
C 1968.
C 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 CHKPRM, SPELIP
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 SEPELI
C
DIMENSION GRHS(IDMN,*) ,USOL(IDMN,*)
DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) ,
1 W(*)
EXTERNAL COFX ,COFY
C***FIRST EXECUTABLE STATEMENT SEPELI
CALL CHKPRM (INTL,IORDER,A,B,M,MBDCND,C,D,N,NBDCND,COFX,COFY,
1 IDMN,IERROR)
IF (IERROR .NE. 0) RETURN
C
C COMPUTE MINIMUM WORK SPACE AND CHECK WORK SPACE LENGTH INPUT
C
L = N+1
IF (NBDCND .EQ. 0) L = N
LOGB2N = INT(LOG(L+0.5)/LOG(2.0))+1
LL = 2**(LOGB2N+1)
K = M+1
L = N+1
LENGTH = (LOGB2N-2)*LL+LOGB2N+MAX(2*L,6*K)+5
IF (NBDCND .EQ. 0) LENGTH = LENGTH+2*L
IERROR = 11
LINPUT = INT(W(1)+0.5)
LOUTPT = LENGTH+6*(K+L)+1
W(1) = LOUTPT
IF (LOUTPT .GT. LINPUT) RETURN
IERROR = 0
C
C SET WORK SPACE INDICES
C
I1 = LENGTH+2
I2 = I1+L
I3 = I2+L
I4 = I3+L
I5 = I4+L
I6 = I5+L
I7 = I6+L
I8 = I7+K
I9 = I8+K
I10 = I9+K
I11 = I10+K
I12 = I11+K
I13 = 2
CALL SPELIP (INTL,IORDER,A,B,M,MBDCND,BDA,ALPHA,BDB,BETA,C,D,N,
1 NBDCND,BDC,GAMA,BDD,XNU,COFX,COFY,W(I1),W(I2),W(I3),
2 W(I4),W(I5),W(I6),W(I7),W(I8),W(I9),W(I10),W(I11),
3 W(I12),GRHS,USOL,IDMN,W(I13),PERTRB,IERROR)
RETURN
END