SUBROUTINE HWSPLR (A,B,M,MBDCND,BDA,BDB,C,D,N,NBDCND,BDC,BDD, 1 ELMBDA,F,IDIMF,PERTRB,IERROR,W) 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 4.0 , JUNE 1999) * 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 C C DIMENSION OF BDA(N),BDB(N),BDC(M),BDD(M),F(IDIMF,N+1), C ARGUMENTS W(SEE ARGUMENT LIST) C C LATEST REVISION NOVEMBER 1988 C C PURPOSE SOLVES A FINITE DIFFERENCE APPROXIMATION TO C THE HELMHOLTZ EQUATION IN POLAR COORDINATES. C THE EQUATION IS C C (1/R)(D/DR)(R(DU/DR)) + C (1/R**2)(D/DTHETA)(DU/DTHETA) + C LAMBDA*U = F(R,THETA). C C USAGE CALL HWSPLR (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, C WHERE DR = (B-A)/M IS THE PANEL WIDTH. C M MUST BE GREATER THAN 3. C C MBDCND C INDICATES THE TYPE OF BOUNDARY CONDITION 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 C THE 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 NOTE: C IF A = 0, DO NOT USE MBDCND = 3 OR 4, BUT C 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,THETA(J)), C 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 OF C THE SOLUTION WITH RESPECT TO R AT R = B. C C WHEN MBDCND = 2,3, OR 6, C BDB(J) = (D/DR)U(B,THETA(J)), C 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 THETA, I.E., C .LE. C THETA .LE. D. 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 THETA-DIRECTION GIVEN BY C THETA(J) = C+(J-1)DTHETA FOR C J = 1,2,...,N+1, WHERE C DTHETA = (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 THETA = C AND AT THETA = D. C C = 0 IF THE SOLUTION IS PERIODIC IN THETA, C I.E., U(I,J) = U(I,N+J). C = 1 IF THE SOLUTION IS SPECIFIED AT C THETA = C AND THETA = D C (SEE NOTE BELOW). C = 2 IF THE SOLUTION IS SPECIFIED AT C THETA = C AND THE DERIVATIVE OF THE C SOLUTION WITH RESPECT TO THETA IS C SPECIFIED AT THETA = D C (SEE NOTE BELOW). C = 4 IF THE DERIVATIVE OF THE SOLUTION C WITH RESPECT TO THETA IS SPECIFIED C AT THETA = C AND THE SOLUTION IS C SPECIFIED AT THETA = D C (SEE NOTE BELOW). C C NOTE: C WHEN NBDCND = 1,2, OR 4, DO NOT USE C MBDCND = 5 OR 6 C (THE FORMER INDICATES THAT THE SOLUTION C IS SPECIFIED AT R = 0, THE LATTER INDICATES C THE SOLUTION IS UNSPECIFIED AT R = 0). C USE INSTEAD MBDCND = 1 OR 2 . 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 THETA AT C THETA = C. WHEN NBDCND = 3 OR 4, C C BDC(I) = (D/DTHETA)U(R(I),C), 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 C OF THE SOLUTION WITH RESPECT TO THETA AT C THETA = D. WHEN NBDCND = 2 OR 3, C C BDD(I) = (D/DTHETA)U(R(I),D), C 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 .LT. 0, A SOLUTION C MAY NOT EXIST. HOWEVER, HWSPLR WILL C ATTEMPT TO FIND A SOLUTION. 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),THETA(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,THETA(J)) U(B,THETA(J)) C 2 U(A,THETA(J)) F(B,THETA(J)) C 3 F(A,THETA(J)) F(B,THETA(J)) C 4 F(A,THETA(J)) U(B,THETA(J)) C 5 F(0,0) U(B,THETA(J)) C 6 F(0,0) F(B,THETA(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 THEN 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 HWSPLR. THIS PARAMETER IS USED TO SPECIFY C THE VARIABLE DIMENSION OF F. IDIMF MUST C BE AT LEAST M+1. C C W C A ONE-DIMENSIONAL ARRAY THAT MUST BE C PROVIDED BY THE USER FOR WORK SPACE. C W MAY REQUIRE UP TO 4*(N+1) + C (13 + INT(LOG2(N+1)))*(M+1) LOCATIONS. C THE ACTUAL NUMBER OF LOCATIONS USED IS C COMPUTED BY HWSPLR AND IS RETURNED IN C LOCATION W(I). 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),THETA(J)), C I = 1,2,...,M+1, J = 1,2,...,N+1 . 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. HWSPLR 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. PERTRB SHOULD BE SMALL COMPARED C TO THE RIGHT SIDE. OTHERWISE, A SOLUTION C IS OBTAINED TO AN ESSENTIALLY DIFFERENT C PROBLEM. THIS COMPARISON SHOULD ALWAYS C BE MADE TO INSURE THAT A MEANINGFUL C SOLUTION HAS BEEN OBTAINED. C C IERROR C AN ERROR FLAG THAT 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 .GT. 4 . C = 7 A = 0, MBDCND = 3 OR 4 . C = 8 A .GT. 0, MBDCND .GE. 5 . C = 9 MBDCND .GE. 5, NBDCND .NE. 0 C AND NBDCND .NE. 3 . C = 10 IDIMF .LT. M+1 . C = 11 LAMBDA .GT. 0 . C = 12 M .LE. 3 C C SINCE THIS IS THE ONLY MEANS OF INDICATING C A POSSIBLY INCORRECT CALL TO HWSPLR, THE C USER SHOULD TEST IERROR AFTER THE CALL. C C W C W(1) CONTAINS THE REQUIRED LENGTH OF W. C C SPECIAL CONDITIONS NONE C C I/O NONE C C PRECISION SINGLE C C REQUIRED LIBRARY GENBUN, GNBNAUX, AND COMF C FILES FROM FISHPAK C C LANGUAGE FORTRAN 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 C PORTABILITY FORTRAN 77 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. C***********************************************************************