c c file hstcyl.txt (documentation for the FISHPACK solver HSTCYL) 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 HSTCYL (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, C + ELMBDA,F,IDIMF,PERTRB,IERROR) C C DIMENSION OF BDA(N),BDB(N),BDC(M),BDD(M),F(IDIMF,N) C ARGUMENTS C C LATEST REVISION June 2004 C C PURPOSE SOLVES THE STANDARD FIVE-POINT FINITE C DIFFERENCE APPROXIMATION ON A STAGGERED C GRID TO THE MODIFIED HELMHOLTZ EQUATION C IN CYLINDRICAL COORDINATES. THIS EQUATION C C (1/R)(D/DR)(R(DU/DR)) + (D/DZ)(DU/DZ) C C + LAMBDA*(1/R**2)*U = F(R,Z) C C IS A TWO-DIMENSIONAL MODIFIED HELMHOLTZ C EQUATION RESULTING FROM THE FOURIER TRANSFORM C OF A THREE-DIMENSIONAL POISSON EQUATION. C C USAGE CALL HSTCYL (A,B,M,MBDCND,BDA,BDB,C,D,N, C NBDCND,BDC,BDD,ELMBDA,F,IDIMF, C PERTRB,IERROR) C C ARGUMENTS C ON INPUT A,B C C THE RANGE OF R, I.E. A .LE. R .LE. B. C A MUST BE LESS THAN B AND A MUST BE C BE NON-NEGATIVE. C C M C THE NUMBER OF GRID POINTS IN THE INTERVAL C (A,B). THE GRID POINTS IN THE R-DIRECTION C R-DIRECTION ARE GIVEN BY C R(I) = A + (I-0.5)DR FOR I=1,2,...,M C WHERE DR =(B-A)/M. C M MUST BE GREATER THAN 2. 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 R = A C (SEE NOTE BELOW) AND R = B. C C = 2 IF THE SOLUTION IS SPECIFIED AT R = A C (SEE NOTE BELOW) AND THE DERIVATIVE C OF THE SOLUTION WITH RESPECT TO R IS C SPECIFIED AT R = B. C 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 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 C = 5 IF THE SOLUTION IS UNSPECIFIED AT C R = A = 0 AND THE SOLUTION IS C SPECIFIED AT R = B. C 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 NOTE: C IF A = 0, DO NOT USE MBDCND = 1,2,3, OR 4, C BUT INSTEAD USE MBDCND = 5 OR 6. C THE RESULTING APPROXIMATION GIVES THE ONLY C MEANINGFUL BOUNDARY CONDITION, C I.E. DU/DR = 0. C (SEE D. GREENSPAN, 'INTRODUCTORY NUMERICAL C ANALYSIS OF ELLIPTIC BOUNDARY VALUE C PROBLEMS,' HARPER AND ROW, 1965, CHAPTER 5.) C C BDA C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT C SPECIFIES THE BOUNDARY VALUES (IF ANY) C OF THE SOLUTION AT R = A. C C WHEN MBDCND = 1 OR 2, C BDA(J) = U(A,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 3 OR 4, C BDA(J) = (D/DR)U(A,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 5 OR 6, BDA IS A DUMMY C VARIABLE. C C BDB C A ONE-DIMENSIONAL ARRAY OF LENGTH N THAT C SPECIFIES THE BOUNDARY VALUES OF THE C SOLUTION AT R = B. C C WHEN MBDCND = 1,4,OR 5, C BDB(J) = U(B,Z(J)) , J=1,2,...,N. C C WHEN MBDCND = 2,3, OR 6, C BDB(J) = (D/DR)U(B,Z(J)) , J=1,2,...,N. 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 UNKNOWNS IN THE INTERVAL C (C,D). THE UNKNOWNS IN THE Z-DIRECTION C ARE GIVEN BY Z(J) = C + (J-0.5)DZ, C J=1,2,...,N, WHERE DZ = (D-C)/N. C N MUST BE GREATER THAN 2. 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, I.E. C U(I,J) = U(I,N+J). C C = 1 IF THE SOLUTION IS SPECIFIED AT Z = C C AND Z = D. C C = 2 IF THE SOLUTION IS SPECIFIED AT Z = C C AND THE DERIVATIVE OF THE SOLUTION WITH C RESPECT TO Z IS SPECIFIED AT Z = D. C C = 3 IF THE DERIVATIVE OF THE SOLUTION WITH C RESPECT TO Z IS SPECIFIED AT Z = C C AND Z = D. C C = 4 IF THE DERIVATIVE OF THE SOLUTION WITH C RESPECT TO Z IS SPECIFIED AT Z = C AND C THE SOLUTION IS SPECIFIED AT Z = D. C C BDC C A ONE DIMENSIONAL ARRAY OF LENGTH M THAT C SPECIFIES THE BOUNDARY VALUES OF THE C SOLUTION AT Z = C. C C WHEN NBDCND = 1 OR 2, C BDC(I) = U(R(I),C) , I=1,2,...,M. C C WHEN NBDCND = 3 OR 4, C BDC(I) = (D/DZ)U(R(I),C), I=1,2,...,M. C C WHEN NBDCND = 0, BDC IS A DUMMY VARIABLE. C C BDD C A ONE-DIMENSIONAL ARRAY OF LENGTH M THAT C SPECIFIES THE BOUNDARY VALUES OF THE C SOLUTION AT Z = D. C C WHEN NBDCND = 1 OR 4, C BDD(I) = U(R(I),D) , I=1,2,...,M. C C WHEN NBDCND = 2 OR 3, C BDD(I) = (D/DZ)U(R(I),D) , I=1,2,...,M. C C WHEN NBDCND = 0, BDD IS A DUMMY VARIABLE. C C ELMBDA C THE CONSTANT LAMBDA IN THE MODIFIED C HELMHOLTZ EQUATION. IF LAMBDA IS GREATER C THAN 0, A SOLUTION MAY NOT EXIST. C HOWEVER, HSTCYL WILL ATTEMPT TO FIND A C SOLUTION. LAMBDA MUST BE ZERO WHEN C MBDCND = 5 OR 6. C C F C A TWO-DIMENSIONAL ARRAY THAT SPECIFIES C THE VALUES OF THE RIGHT SIDE OF THE C MODIFIED HELMHOLTZ EQUATION. C FOR I=1,2,...,M AND J=1,2,...,N C F(I,J) = F(R(I),Z(J)) . C F MUST BE DIMENSIONED AT LEAST M X N. C C IDIMF C THE ROW (OR FIRST) DIMENSION OF THE ARRAY C F AS IT APPEARS IN THE PROGRAM CALLING C HSTCYL. THIS PARAMETER IS USED TO SPECIFY C THE VARIABLE DIMENSION OF F. IDIMF MUST C BE AT LEAST M. C C ON OUTPUT C C F C CONTAINS THE SOLUTION U(I,J) OF THE FINITE C DIFFERENCE APPROXIMATION FOR THE GRID POINT C (R(I),Z(J)) FOR I=1,2,...,M, J=1,2,...,N. C C PERTRB C IF A COMBINATION OF PERIODIC, DERIVATIVE, C OR UNSPECIFIED BOUNDARY CONDITIONS IS C SPECIFIED 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. HSTCYL 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. C THIS COMPARISON SHOULD ALWAYS BE MADE TO C INSURE THAT A MEANINGFUL SOLUTION HAS BEEN C OBTAINED. C C IERROR C AN ERROR FLAG THAT INDICATES INVALID INPUT C PARAMETERS. EXCEPT TO NUMBERS 0 AND 11, C A SOLUTION IS NOT ATTEMPTED. C C = 0 NO ERROR C C = 1 A .LT. 0 C C = 2 A .GE. B C C = 3 MBDCND .LT. 1 OR MBDCND .GT. 6 C C = 4 C .GE. D C C = 5 N .LE. 2 C C = 6 NBDCND .LT. 0 OR NBDCND .GT. 4 C C = 7 A = 0 AND MBDCND = 1,2,3, OR 4 C C = 8 A .GT. 0 AND MBDCND .GE. 5 C C = 9 M .LE. 2 C C = 10 IDIMF .LT. M C C = 11 LAMBDA .GT. 0 C C = 12 A=0, MBDCND .GE. 5, ELMBDA .NE. 0 C C SINCE THIS IS THE ONLY MEANS OF INDICATING C A POSSIBLY INCORRECT CALL TO HSTCYL, THE C USER SHOULD TEST IERROR AFTER THE CALL. C 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 I/O NONE C C PRECISION SINGLE C C REQUIRED LIBRARY fish.f,comf.f,genbun.f,gnbnaux.f,poistg.f C FILES C C LANGUAGE FORTRAN 90 C C HISTORY WRITTEN BY ROLAND SWEET AT NCAR IN 1977. C RELEASED ON NCAR'S PUBLIC SOFTWARE LIBRARIES C IN JANUARY 1980. c Revised in June 2004 by John Adams using c Fortran 90 dynamically allocated work space. C C PORTABILITY FORTRAN 90 C C ALGORITHM THIS SUBROUTINE DEFINES THE FINITE-DIFFERENCE C EQUATIONS, INCORPORATES BOUNDARY DATA, ADJUSTS C THE RIGHT SIDE WHEN THE SYSTEM IS SINGULAR AND C CALLS EITHER POISTG OR GENBUN WHICH SOLVES THE C LINEAR SYSTEM OF EQUATIONS. C C TIMING FOR LARGE M AND N, THE OPERATION COUNT C IS ROUGHLY PROPORTIONAL TO M*N*LOG2(N). C C ACCURACY THE SOLUTION PROCESS RESULTS IN A LOSS C OF NO MORE THAN FOUR SIGNIFICANT DIGITS C FOR N AND M AS LARGE AS 64. C MORE DETAILED INFORMATION ABOUT ACCURACY C CAN BE FOUND IN THE DOCUMENTATION FOR C SUBROUTINE POISTG WHICH IS THE ROUTINE THAT C ACTUALLY SOLVES THE FINITE DIFFERENCE C EQUATIONS. C C REFERENCES U. SCHUMANN AND R. SWEET, "A DIRECT METHOD FOR C THE SOLUTION OF POISSON'S EQUATION WITH NEUMANN C BOUNDARY CONDITIONS ON A STAGGERED GRID OF C ARBITRARY SIZE," J. COMP. PHYS. 20(1976), C PP. 171-182. C***********************************************************************