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451 lines
21 KiB
Fortran
451 lines
21 KiB
Fortran
*DECK SEPX4
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SUBROUTINE SEPX4 (IORDER, A, B, M, MBDCND, BDA, ALPHA, BDB, BETA,
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+ C, D, N, NBDCND, BDC, BDD, COFX, GRHS, USOL, IDMN, W, PERTRB,
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+ IERROR)
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C***BEGIN PROLOGUE SEPX4
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C***PURPOSE Solve for either the second or fourth order finite
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C difference approximation to the solution of a separable
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C elliptic partial differential equation on a rectangle.
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C Any combination of periodic or mixed boundary conditions is
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C allowed.
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C***LIBRARY SLATEC (FISHPACK)
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C***CATEGORY I2B1A2
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C***TYPE SINGLE PRECISION (SEPX4-S)
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C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, SEPARABLE
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C***AUTHOR Adams, J., (NCAR)
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C Swarztrauber, P. N., (NCAR)
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C Sweet, R., (NCAR)
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C***DESCRIPTION
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C
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C Purpose SEPX4 solves for either the second-order
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C finite difference approximation or a
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C fourth-order approximation to the
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C solution of a separable elliptic equation
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C AF(X)*UXX+BF(X)*UX+CF(X)*U+UYY = G(X,Y)
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C
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C on a rectangle (X greater than or equal to A
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C and less than or equal to B; Y greater than
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C or equal to C and less than or equal to D).
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C Any combination of periodic or mixed boundary
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C conditions is allowed.
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C If boundary conditions in the X direction
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C are periodic (see MBDCND=0 below) then the
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C coefficients must satisfy
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C AF(X)=C1,BF(X)=0,CF(X)=C2 for all X.
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C Here C1,C2 are constants, C1.GT.0.
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C
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C The possible boundary conditions are
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C in the X-direction:
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C (0) Periodic, U(X+B-A,Y)=U(X,Y) for all Y,X
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C (1) U(A,Y), U(B,Y) are specified for all Y
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C (2) U(A,Y), dU(B,Y)/dX+BETA*U(B,Y) are
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C specified for all Y
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C (3) dU(A,Y)/dX+ALPHA*U(A,Y),dU(B,Y)/dX+
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C BETA*U(B,Y) are specified for all Y
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C (4) dU(A,Y)/dX+ALPHA*U(A,Y),U(B,Y) are
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C specified for all Y
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C
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C In the Y-direction:
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C (0) Periodic, U(X,Y+D-C)=U(X,Y) for all X,Y
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C (1) U(X,C),U(X,D) are specified for all X
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C (2) U(X,C),dU(X,D)/dY are specified for all X
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C (3) dU(X,C)/DY,dU(X,D)/dY are specified for
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C all X
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C (4) dU(X,C)/DY,U(X,D) are specified for all X
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C
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C Usage Call SEPX4(IORDER,A,B,M,MBDCND,BDA,ALPHA,BDB,
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C BETA,C,D,N,NBDCND,BDC,BDD,COFX,
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C GRHS,USOL,IDMN,W,PERTRB,IERROR)
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C
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C Arguments
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C
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C IORDER
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C = 2 If a second-order approximation is sought
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C = 4 If a fourth-order approximation is sought
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C
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C A,B
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C The range of the X-independent variable;
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C i.e., X is greater than or equal to A and
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C less than or equal to B. A must be less than
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C B.
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C
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C M
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C The number of panels into which the interval
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C [A,B] is subdivided. Hence, there will be
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C M+1 grid points in the X-direction given by
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C XI=A+(I-1)*DLX for I=1,2,...,M+1 where
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C DLX=(B-A)/M is the panel width. M must be
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C less than IDMN and greater than 5.
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C
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C MBDCND
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C Indicates the type of boundary condition at
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C X=A and X=B
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C = 0 If the solution is periodic in X; i.e.,
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C U(X+B-A,Y)=U(X,Y) for all Y,X
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C = 1 If the solution is specified at X=A and
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C X=B; i.e., U(A,Y) and U(B,Y) are
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C specified for all Y
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C = 2 If the solution is specified at X=A and
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C the boundary condition is mixed at X=B;
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C i.e., U(A,Y) and dU(B,Y)/dX+BETA*U(B,Y)
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C are specified for all Y
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C = 3 If the boundary conditions at X=A and X=B
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C are mixed; i.e., dU(A,Y)/dX+ALPHA*U(A,Y)
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C and dU(B,Y)/dX+BETA*U(B,Y) are specified
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C for all Y
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C = 4 If the boundary condition at X=A is mixed
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C and the solution is specified at X=B;
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C i.e., dU(A,Y)/dX+ALPHA*U(A,Y) and U(B,Y)
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C are specified for all Y
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C
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C BDA
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C A one-dimensional array of length N+1 that
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C specifies the values of dU(A,Y)/dX+
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C ALPHA*U(A,Y) at X=A, when MBDCND=3 or 4.
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C BDA(J) = dU(A,YJ)/dX+ALPHA*U(A,YJ);
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C J=1,2,...,N+1
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C When MBDCND has any other value, BDA is a
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C dummy parameter.
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C
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C On Input ALPHA
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C The scalar multiplying the solution in case
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C of a mixed boundary condition AT X=A (see
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C argument BDA). If MBDCND = 3,4 then ALPHA is
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C a dummy parameter.
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C
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C BDB
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C A one-dimensional array of length N+1 that
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C specifies the values of dU(B,Y)/dX+
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C BETA*U(B,Y) at X=B. when MBDCND=2 or 3
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C BDB(J) = dU(B,YJ)/dX+BETA*U(B,YJ);
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C J=1,2,...,N+1
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C When MBDCND has any other value, BDB is a
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C dummy parameter.
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C
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C BETA
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C The scalar multiplying the solution in case
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C of a mixed boundary condition at X=B (see
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C argument BDB). If MBDCND=2,3 then BETA is a
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C dummy parameter.
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C
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C C,D
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C The range of the Y-independent variable;
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C i.e., Y is greater than or equal to C and
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C less than or equal to D. C must be less than
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C D.
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C
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C N
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C The number of panels into which the interval
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C [C,D] is subdivided. Hence, there will be
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C N+1 grid points in the Y-direction given by
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C YJ=C+(J-1)*DLY for J=1,2,...,N+1 where
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C DLY=(D-C)/N is the panel width. In addition,
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C N must be greater than 4.
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C
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C NBDCND
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C Indicates the types of boundary conditions at
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C Y=C and Y=D
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C = 0 If the solution is periodic in Y; i.e.,
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C U(X,Y+D-C)=U(X,Y) for all X,Y
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C = 1 If the solution is specified at Y=C and
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C Y = D, i.e., U(X,C) and U(X,D) are
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C specified for all X
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C = 2 If the solution is specified at Y=C and
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C the boundary condition is mixed at Y=D;
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C i.e., dU(X,C)/dY and U(X,D)
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C are specified for all X
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C = 3 If the boundary conditions are mixed at
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C Y= C and Y=D i.e., dU(X,D)/DY
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C and dU(X,D)/dY are specified
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C for all X
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C = 4 If the boundary condition is mixed at Y=C
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C and the solution is specified at Y=D;
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C i.e. dU(X,C)/dY+GAMA*U(X,C) and U(X,D)
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C are specified for all X
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C
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C BDC
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C A one-dimensional array of length M+1 that
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C specifies the value dU(X,C)/DY
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C at Y=C. When NBDCND=3 or 4
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C BDC(I) = dU(XI,C)/DY
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C I=1,2,...,M+1.
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C When NBDCND has any other value, BDC is a
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C dummy parameter.
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C
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C
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C BDD
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C A one-dimensional array of length M+1 that
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C specifies the value of dU(X,D)/DY
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C at Y=D. When NBDCND=2 or 3
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C BDD(I)=dU(XI,D)/DY
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C I=1,2,...,M+1.
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C When NBDCND has any other value, BDD is a
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C dummy parameter.
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C
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C
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C COFX
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C A user-supplied subprogram with
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C parameters X, AFUN, BFUN, CFUN which
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C returns the values of the X-dependent
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C coefficients AF(X), BF(X), CF(X) in
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C the elliptic equation at X.
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C If boundary conditions in the X direction
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C are periodic then the coefficients
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C must satisfy AF(X)=C1,BF(X)=0,CF(X)=C2 for
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C all X. Here C1.GT.0 and C2 are constants.
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C
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C Note that COFX must be declared external
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C in the calling routine.
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C
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C GRHS
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C A two-dimensional array that specifies the
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C values of the right-hand side of the elliptic
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C equation; i.e., GRHS(I,J)=G(XI,YI), for
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C I=2,...,M; J=2,...,N. At the boundaries,
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C GRHS is defined by
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C
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C MBDCND GRHS(1,J) GRHS(M+1,J)
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C ------ --------- -----------
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C 0 G(A,YJ) G(B,YJ)
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C 1 * *
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C 2 * G(B,YJ) J=1,2,...,N+1
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C 3 G(A,YJ) G(B,YJ)
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C 4 G(A,YJ) *
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C
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C NBDCND GRHS(I,1) GRHS(I,N+1)
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C ------ --------- -----------
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C 0 G(XI,C) G(XI,D)
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C 1 * *
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C 2 * G(XI,D) I=1,2,...,M+1
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C 3 G(XI,C) G(XI,D)
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C 4 G(XI,C) *
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C
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C where * means these quantities are not used.
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C GRHS should be dimensioned IDMN by at least
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C N+1 in the calling routine.
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C
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C USOL
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C A two-dimensional array that specifies the
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C values of the solution along the boundaries.
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C At the boundaries, USOL is defined by
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C
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C MBDCND USOL(1,J) USOL(M+1,J)
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C ------ --------- -----------
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C 0 * *
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C 1 U(A,YJ) U(B,YJ)
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C 2 U(A,YJ) * J=1,2,...,N+1
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C 3 * *
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C 4 * U(B,YJ)
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C
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C NBDCND USOL(I,1) USOL(I,N+1)
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C ------ --------- -----------
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C 0 * *
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C 1 U(XI,C) U(XI,D)
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C 2 U(XI,C) * I=1,2,...,M+1
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C 3 * *
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C 4 * U(XI,D)
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C
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C where * means the quantities are not used in
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C the solution.
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C
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C If IORDER=2, the user may equivalence GRHS
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C and USOL to save space. Note that in this
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C case the tables specifying the boundaries of
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C the GRHS and USOL arrays determine the
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C boundaries uniquely except at the corners.
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C If the tables call for both G(X,Y) and
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C U(X,Y) at a corner then the solution must be
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C chosen. For example, if MBDCND=2 and
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C NBDCND=4, then U(A,C), U(A,D), U(B,D) must be
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C chosen at the corners in addition to G(B,C).
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C
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C If IORDER=4, then the two arrays, USOL and
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C GRHS, must be distinct.
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C
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C USOL should be dimensioned IDMN by at least
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C N+1 in the calling routine.
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C
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C IDMN
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C The row (or first) dimension of the arrays
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C GRHS and USOL as it appears in the program
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C calling SEPX4. This parameter is used to
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C specify the variable dimension of GRHS and
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C USOL. IDMN must be at least 7 and greater
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C than or equal to M+1.
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C
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C W
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C A one-dimensional array that must be provided
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C by the user for work space.
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C 10*N+(16+INT(log2(N)))*(M+1)+23 will suffice
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C as a length for W. The actual length of
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C W in the calling routine must be set in W(1)
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C (see IERROR=11).
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C
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C On Output USOL
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C Contains the approximate solution to the
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C elliptic equation. USOL(I,J) is the
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C approximation to U(XI,YJ) for I=1,2...,M+1
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C and J=1,2,...,N+1. The approximation has
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C error O(DLX**2+DLY**2) if called with
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C IORDER=2 and O(DLX**4+DLY**4) if called with
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C IORDER=4.
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C
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C W
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C W(1) contains the exact minimal length (in
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C floating point) required for the work space
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C (see IERROR=11).
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C
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C PERTRB
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C If a combination of periodic or derivative
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C boundary conditions (i.e., ALPHA=BETA=0 if
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C MBDCND=3) is specified and if CF(X)=0 for all
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C X, then a solution to the discretized matrix
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C equation may not exist (reflecting the non-
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C uniqueness of solutions to the PDE). PERTRB
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C is a constant calculated and subtracted from
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C the right hand side of the matrix equation
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C insuring the existence of a solution.
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C SEPX4 computes this solution which is a
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C weighted minimal least squares solution to
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C the original problem. If singularity is
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C not detected PERTRB=0.0 is returned by
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C SEPX4.
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C
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C IERROR
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C An error flag that indicates invalid input
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C parameters or failure to find a solution
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C = 0 No error
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C = 1 If A greater than B or C greater than D
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C = 2 If MBDCND less than 0 or MBDCND greater
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C than 4
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C = 3 If NBDCND less than 0 or NBDCND greater
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C than 4
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C = 4 If attempt to find a solution fails.
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C (the linear system generated is not
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C diagonally dominant.)
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C = 5 If IDMN is too small (see discussion of
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C IDMN)
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C = 6 If M is too small or too large (see
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C discussion of M)
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C = 7 If N is too small (see discussion of N)
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C = 8 If IORDER is not 2 or 4
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C = 10 If AFUN is less than or equal to zero
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C for some interior mesh point XI
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C = 11 If the work space length input in W(1)
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C is less than the exact minimal work
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C space length required output in W(1).
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C = 12 If MBDCND=0 and AF(X)=CF(X)=constant
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C or BF(X)=0 for all X is not true.
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C
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C *Long Description:
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C
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C Dimension of BDA(N+1), BDB(N+1), BDC(M+1), BDD(M+1),
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C Arguments USOL(IDMN,N+1), GRHS(IDMN,N+1),
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C W (see argument list)
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C
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C Latest Revision October 1980
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C
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C Special Conditions NONE
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C
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C Common Blocks SPL4
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C
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C I/O NONE
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C
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C Precision Single
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C
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C Required Library NONE
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C Files
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C
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C Specialist John C. Adams, NCAR, Boulder, Colorado 80307
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C
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C Language FORTRAN
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C
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C
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C Entry Points SEPX4,SPELI4,CHKPR4,CHKSN4,ORTHO4,MINSO4,TRIS4,
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C DEFE4,DX4,DY4
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C
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C History SEPX4 was developed by modifying the ULIB
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C routine SEPELI during October 1978.
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C It should be used instead of SEPELI whenever
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C possible. The increase in speed is at least
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C a factor of three.
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C
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C Algorithm SEPX4 automatically discretizes the separable
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C elliptic equation which is then solved by a
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C generalized cyclic reduction algorithm in the
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C subroutine POIS. The fourth order solution
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C is obtained using the technique of
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C deferred corrections referenced below.
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C
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C
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C References Keller, H.B., 'Numerical Methods for Two-point
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C Boundary-value Problems', Blaisdel (1968),
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C Waltham, Mass.
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C
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C Swarztrauber, P., and R. Sweet (1975):
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C 'Efficient FORTRAN Subprograms For The
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C Solution of Elliptic Partial Differential
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C Equations'. NCAR Technical Note
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C NCAR-TN/IA-109, pp. 135-137.
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C
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C***REFERENCES H. B. Keller, Numerical Methods for Two-point
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C Boundary-value Problems, Blaisdel, Waltham, Mass.,
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C 1968.
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C P. N. Swarztrauber and R. Sweet, Efficient Fortran
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C subprograms for the solution of elliptic equations,
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C NCAR TN/IA-109, July 1975, 138 pp.
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C***ROUTINES CALLED CHKPR4, SPELI4
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C***REVISION HISTORY (YYMMDD)
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C 801001 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890531 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 920122 Minor corrections and modifications to prologue. (WRB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE SEPX4
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C
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DIMENSION GRHS(IDMN,*) ,USOL(IDMN,*)
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DIMENSION BDA(*) ,BDB(*) ,BDC(*) ,BDD(*) ,
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1 W(*)
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EXTERNAL COFX
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C***FIRST EXECUTABLE STATEMENT SEPX4
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CALL CHKPR4(IORDER,A,B,M,MBDCND,C,D,N,NBDCND,COFX,IDMN,IERROR)
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IF (IERROR .NE. 0) RETURN
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C
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C COMPUTE MINIMUM WORK SPACE AND CHECK WORK SPACE LENGTH INPUT
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C
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L = N+1
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IF (NBDCND .EQ. 0) L = N
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K = M+1
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L = N+1
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C ESTIMATE LOG BASE 2 OF N
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LOG2N=INT(LOG(REAL(N+1))/LOG(2.0)+0.5)
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LENGTH=4*(N+1)+(10+LOG2N)*(M+1)
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IERROR = 11
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LINPUT = INT(W(1)+0.5)
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LOUTPT = LENGTH+6*(K+L)+1
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W(1) = LOUTPT
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IF (LOUTPT .GT. LINPUT) RETURN
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IERROR = 0
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C
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C SET WORK SPACE INDICES
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C
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I1 = LENGTH+2
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I2 = I1+L
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I3 = I2+L
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I4 = I3+L
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I5 = I4+L
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I6 = I5+L
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I7 = I6+L
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I8 = I7+K
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I9 = I8+K
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I10 = I9+K
|
|
I11 = I10+K
|
|
I12 = I11+K
|
|
I13 = 2
|
|
CALL SPELI4(IORDER,A,B,M,MBDCND,BDA,ALPHA,BDB,BETA,C,D,N,
|
|
1NBDCND,BDC,BDD,COFX,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
|