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- *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
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