c c file hwscyl.txt (documentation for the FISHPACK solver HWSCYL) c c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . . c . copyright (c) 2004 by UCAR . c . . c . UNIVERSITY CORPORATION for ATMOSPHERIC RESEARCH . c . . c . all rights reserved . c . . c . . c . FISHPACK version 5.0 . c . . c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C * * C * F I S H P A C K * C * * C * * C * A PACKAGE OF FORTRAN SUBPROGRAMS FOR THE SOLUTION OF * C * * C * SEPARABLE ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS * C * * C * (Version 5.0 , JUNE 2004) * C * * C * BY * C * * C * JOHN ADAMS, PAUL SWARZTRAUBER AND ROLAND SWEET * C * * C * OF * C * * C * THE NATIONAL CENTER FOR ATMOSPHERIC RESEARCH * C * * C * BOULDER, COLORADO (80307) U.S.A. * C * * C * WHICH IS SPONSORED BY * C * * C * THE NATIONAL SCIENCE FOUNDATION * C * * C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C SUBROUTINE HWSCYL (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, C + ELMBDA,F,IDIMF,PERTRB,IERROR) C C C DIMENSION OF BDA(N),BDB(N),BDC(M),BDD(M),F(IDIMF,N+1) C ARGUMENTS C C LATEST REVISION June 2004 C C PURPOSE SOLVES A FINITE DIFFERENCE APPROXIMATION C TO THE HELMHOLTZ EQUATION IN CYLINDRICAL C COORDINATES. THIS MODIFIED HELMHOLTZ EQUATION C C (1/R)(D/DR)(R(DU/DR)) + (D/DZ)(DU/DZ) C C + (LAMBDA/R**2)U = F(R,Z) C C RESULTS FROM THE FOURIER TRANSFORM OF THE C THREE-DIMENSIONAL POISSON EQUATION. C C USAGE CALL HWSCYL (A,B,M,MBDCND,BDA,BDB,C,D,N, C NBDCND,BDC,BDD,ELMBDA,F,IDIMF, C PERTRB,IERROR,W) C C ARGUMENTS C ON INPUT A,B C THE RANGE OF R, I.E., A .LE. R .LE. B. C A MUST BE LESS THAN B AND A MUST BE C NON-NEGATIVE. C C M C THE NUMBER OF PANELS INTO WHICH THE C INTERVAL (A,B) IS SUBDIVIDED. HENCE, C THERE WILL BE M+1 GRID POINTS IN THE C R-DIRECTION GIVEN BY R(I) = A+(I-1)DR, C FOR I = 1,2,...,M+1, WHERE DR = (B-A)/M C IS THE PANEL WIDTH. M MUST BE GREATER C THAN 3. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS C AT R = A AND R = B. C C = 1 IF THE SOLUTION IS SPECIFIED AT C R = A AND R = B. C = 2 IF THE SOLUTION IS SPECIFIED AT C R = A AND THE DERIVATIVE OF THE C SOLUTION WITH RESPECT TO R IS C SPECIFIED AT R = B. C = 3 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO R IS SPECIFIED AT C R = A (SEE NOTE BELOW) AND R = B. C = 4 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO R IS SPECIFIED AT C R = A (SEE NOTE BELOW) AND THE C SOLUTION IS SPECIFIED AT R = B. C = 5 IF THE SOLUTION IS UNSPECIFIED AT C R = A = 0 AND THE SOLUTION IS C SPECIFIED AT R = B. C = 6 IF THE SOLUTION IS UNSPECIFIED AT C R = A = 0 AND THE DERIVATIVE OF THE C SOLUTION WITH RESPECT TO R IS SPECIFIED C AT R = B. C C IF A = 0, DO NOT USE MBDCND = 3 OR 4, C BUT INSTEAD USE MBDCND = 1,2,5, OR 6 . C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO R AT R = A. C C WHEN MBDCND = 3 OR 4, C BDA(J) = (D/DR)U(A,Z(J)), J = 1,2,...,N+1. C C WHEN MBDCND HAS ANY OTHER VALUE, BDA IS C A DUMMY VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO R AT R = B. C C WHEN MBDCND = 2,3, OR 6, C BDB(J) = (D/DR)U(B,Z(J)), J = 1,2,...,N+1. C C WHEN MBDCND HAS ANY OTHER VALUE, BDB IS C A DUMMY VARIABLE. C C C,D C THE RANGE OF Z, I.E., C .LE. Z .LE. D. C C MUST BE LESS THAN D. C C N C THE NUMBER OF PANELS INTO WHICH THE C INTERVAL (C,D) IS SUBDIVIDED. HENCE, C THERE WILL BE N+1 GRID POINTS IN THE C Z-DIRECTION GIVEN BY Z(J) = C+(J-1)DZ, C FOR J = 1,2,...,N+1, C WHERE DZ = (D-C)/N IS THE PANEL WIDTH. C N MUST BE GREATER THAN 3. C C NBDCND C INDICATES THE TYPE OF BOUNDARY CONDITIONS C AT Z = C AND Z = D. C C = 0 IF THE SOLUTION IS PERIODIC IN Z, C I.E., U(I,1) = U(I,N+1). C = 1 IF THE SOLUTION IS SPECIFIED AT C Z = C AND Z = D. C = 2 IF THE SOLUTION IS SPECIFIED AT C Z = C AND THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO Z IS C SPECIFIED AT Z = D. C = 3 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO Z IS C SPECIFIED AT Z = C AND Z = D. C = 4 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO Z IS SPECIFIED AT C Z = C AND THE SOLUTION IS SPECIFIED C AT Z = D. C C BDC C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO Z AT Z = C. C C WHEN NBDCND = 3 OR 4, C BDC(I) = (D/DZ)U(R(I),C), I = 1,2,...,M+1. C C WHEN NBDCND HAS ANY OTHER VALUE, BDC IS C A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT C SPECIFIES THE VALUES OF THE DERIVATIVE OF C THE SOLUTION WITH RESPECT TO Z AT Z = D. C C WHEN NBDCND = 2 OR 3, C BDD(I) = (D/DZ)U(R(I),D), I = 1,2,...,M+1 C C WHEN NBDCND HAS ANY OTHER VALUE, BDD IS C A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE HELMHOLTZ C EQUATION. IF LAMBDA .GT. 0, A SOLUTION C MAY NOT EXIST. HOWEVER, HWSCYL WILL C ATTEMPT TO FIND A SOLUTION. LAMBDA MUST C BE ZERO WHEN MBDCND = 5 OR 6 . C C F C A TWO-DIMENSIONAL ARRAY, OF DIMENSION AT C LEAST (M+1)*(N+1), SPECIFYING VALUES C OF THE RIGHT SIDE OF THE HELMHOLTZ C EQUATION AND BOUNDARY DATA (IF ANY). C C ON THE INTERIOR, F IS DEFINED AS FOLLOWS: C FOR I = 2,3,...,M AND J = 2,3,...,N C F(I,J) = F(R(I),Z(J)). C C ON THE BOUNDARIES F IS DEFINED AS FOLLOWS: C FOR J = 1,2,...,N+1 AND I = 1,2,...,M+1 C C MBDCND F(1,J) F(M+1,J) C ------ --------- --------- C C 1 U(A,Z(J)) U(B,Z(J)) C 2 U(A,Z(J)) F(B,Z(J)) C 3 F(A,Z(J)) F(B,Z(J)) C 4 F(A,Z(J)) U(B,Z(J)) C 5 F(0,Z(J)) U(B,Z(J)) C 6 F(0,Z(J)) F(B,Z(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) C 3 F(R(I),C) F(R(I),D) C 4 F(R(I),C) U(R(I),D) C C NOTE: C IF THE TABLE CALLS FOR BOTH THE SOLUTION C U AND THE RIGHT SIDE F AT A CORNER THEN C THE SOLUTION MUST BE SPECIFIED. C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY C F AS IT APPEARS IN THE PROGRAM CALLING C HWSCYL. THIS PARAMETER IS USED TO SPECIFY C THE VARIABLE DIMENSION OF F. IDIMF MUST C BE AT LEAST M+1 . C C C ON OUTPUT F C CONTAINS THE SOLUTION U(I,J) OF THE FINITE C DIFFERENCE APPROXIMATION FOR THE GRID POINT C (R(I),Z(J)), I =1,2,...,M+1, J =1,2,...,N+1. C C PERTRB C IF ONE SPECIFIES A COMBINATION OF PERIODIC, C DERIVATIVE, AND UNSPECIFIED BOUNDARY C CONDITIONS FOR A POISSON EQUATION C (LAMBDA = 0), A SOLUTION MAY NOT EXIST. C PERTRB IS A CONSTANT, CALCULATED AND C SUBTRACTED FROM F, WHICH ENSURES THAT A C SOLUTION EXISTS. HWSCYL THEN COMPUTES C THIS SOLUTION, WHICH IS A LEAST SQUARES C SOLUTION TO THE ORIGINAL APPROXIMATION. C THIS SOLUTION PLUS ANY CONSTANT IS ALSO C A SOLUTION. HENCE, THE SOLUTION IS NOT C UNIQUE. THE VALUE OF PERTRB SHOULD BE C SMALL COMPARED TO THE RIGHT SIDE F. C OTHERWISE, A SOLUTION IS OBTAINED TO AN C ESSENTIALLY DIFFERENT PROBLEM. THIS C COMPARISON SHOULD ALWAYS BE MADE TO INSURE C THAT A MEANINGFUL SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG WHICH INDICATES INVALID INPUT C PARAMETERS. EXCEPT FOR NUMBERS 0 AND 11, C 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 NBDCND .GT. 4 . C = 7 A = 0, MBDCND = 3 OR 4 . C = 8 A .GT. 0, MBDCND .GE. 5 . C = 9 A = 0, LAMBDA .NE. 0, MBDCND .GE. 5 . C = 10 IDIMF .LT. M+1 . C = 11 LAMBDA .GT. 0 . C = 12 M .LE. 3 C = 20 If the dynamic allocation of real and C complex work space required for solution C fails (for example if N,M are too large C for your computer) C C SINCE THIS IS THE ONLY MEANS OF INDICATING C A POSSIBLY INCORRECT CALL TO HWSCYL, THE C USER SHOULD TEST IERROR AFTER THE CALL. C C C SPECIAL CONDITIONS NONE C C I/O NONE C C PRECISION SINGLE C C REQUIRED files fish.f,genbun.f,gnbnaux.f,comf.f C C LANGUAGE FORTRAN 90 C C HISTORY WRITTEN BY ROLAND SWEET AT NCAR IN THE LATE C 1970'S. RELEASED ON NCAR'S PUBLIC SOFTWARE C LIBRARIES IN JANUARY 1980. c Revised in June 2004 by John Adams using c Fortran 90 dynamically allocated work space. C C C PORTABILITY FORTRAN 90 C C ALGORITHM THE ROUTINE DEFINES THE FINITE DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, AND C ADJUSTS THE RIGHT SIDE OF SINGULAR SYSTEMS C AND THEN CALLS GENBUN TO SOLVE THE SYSTEM. C C TIMING FOR LARGE M AND N, THE OPERATION COUNT C IS ROUGHLY PROPORTIONAL TO C M*N*(LOG2(N) C BUT ALSO DEPENDS ON INPUT PARAMETERS NBDCND C AND MBDCND. C C ACCURACY THE SOLUTION PROCESS EMPLOYED RESULTS IN A LOSS C OF NO MORE THAN THREE SIGNIFICANT DIGITS FOR N C AND M AS LARGE AS 64. MORE DETAILS 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 REFERENCES SWARZTRAUBER,P. AND R. SWEET, "EFFICIENT C FORTRAN SUBPROGRAMS FOR THE SOLUTION OF C ELLIPTIC EQUATIONS" C NCAR TN/IA-109, JULY, 1975, 138 PP. 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