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| 2 | ! |
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| 3 | ! This program may be freely redistributed under the |
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| 4 | ! condition that the copyright notices (including this |
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| 5 | ! entire header) are not removed, and no compensation |
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| 6 | ! is received through use of the software. Private, |
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| 7 | ! research, and institutional use is free. You may |
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| 8 | ! distribute modified versions of this code UNDER THE |
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| 9 | ! CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE |
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| 10 | ! TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE |
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| 11 | ! ORIGINAL AUTHOR, BOTH SOURCE AND OBJECT CODE ARE |
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| 12 | ! MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR |
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| 13 | ! NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution |
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| 14 | ! of this code as part of a commercial system is |
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| 15 | ! permissible ONLY BY DIRECT ARRANGEMENT WITH THE |
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| 16 | ! AUTHOR. (If you are not directly supplying this |
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| 17 | ! code to a customer, and you are instead telling them |
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| 18 | ! how they can obtain it for free, then you are not |
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| 19 | ! required to make any arrangement with me.) |
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| 20 | ! |
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| 21 | ! Disclaimer: Neither I nor: Columbia University, the |
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| 22 | ! National Aeronautics and Space Administration, nor |
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| 23 | ! the Massachusetts Institute of Technology warrant |
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| 24 | ! or certify this code in any way whatsoever. This |
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| 25 | ! code is provided "as-is" to be used at your own risk. |
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| 26 | ! |
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| 27 | ! |
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| 28 | |
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| 29 | ! |
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| 30 | ! P5E.f90: set edge estimates via degree-5 polynomials. |
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| 31 | ! |
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| 32 | ! Darren Engwirda |
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| 33 | ! 25-Mar-2019 |
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| 34 | ! de2363 [at] columbia [dot] edu |
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| 35 | ! |
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| 36 | ! |
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| 37 | |
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| 38 | subroutine p5e(npos,nvar,ndof,delx, & |
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| 39 | & fdat,bclo,bchi,edge, & |
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| 40 | & dfdx,opts,dmin) |
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| 41 | |
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| 42 | ! |
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| 43 | ! NPOS no. edges over grid. |
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| 44 | ! NVAR no. state variables. |
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| 45 | ! NDOF no. degrees-of-freedom per grid-cell. |
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| 46 | ! DELX grid-cell spacing array. LENGTH(DELX) == +1 if |
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| 47 | ! spacing is uniform . |
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| 48 | ! FDAT grid-cell moments array. FDAT is an array with |
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| 49 | ! SIZE = NDOF-by-NVAR-by-NPOS-1 . |
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| 50 | ! BCLO boundary condition at lower endpoint. |
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| 51 | ! BCHI boundary condition at upper endpoint. |
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| 52 | ! EDGE edge-centred interp. for function-value. EDGE |
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| 53 | ! is an array with SIZE = NVAR-by-NPOS . |
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| 54 | ! DFDX edge-centred interp. for 1st-derivative. DFDX |
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| 55 | ! is an array with SIZE = NVAR-by-NPOS . |
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| 56 | ! OPTS method parameters. See RCON-OPTS for details . |
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| 57 | ! DMIN min. grid-cell spacing thresh . |
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| 58 | ! |
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| 59 | |
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| 60 | implicit none |
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| 61 | |
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| 62 | !------------------------------------------- arguments ! |
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| 63 | integer, intent( in) :: npos,nvar,ndof |
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| 64 | real*8 , intent( in) :: delx(:) |
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| 65 | real*8 , intent( in) :: fdat(:,:,:) |
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| 66 | type (rcon_ends), intent(in) :: bclo(:) |
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| 67 | type (rcon_ends), intent(in) :: bchi(:) |
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| 68 | real*8 , intent(out) :: edge(:,:) |
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| 69 | real*8 , intent(out) :: dfdx(:,:) |
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| 70 | real*8 , intent( in) :: dmin |
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| 71 | class(rcon_opts), intent(in) :: opts |
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| 72 | |
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| 73 | !------------------------------------------- variables ! |
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| 74 | integer :: ipos,ivar,idof,head,tail |
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| 75 | logical :: okay |
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| 76 | real*8 :: xhat,fEPS |
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| 77 | real*8 :: delh(-3:+2) |
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| 78 | real*8 :: xmap(-3:+3) |
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| 79 | real*8 :: fhat(+6, nvar) |
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| 80 | real*8 :: ivec(+6,-3:+3) |
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| 81 | real*8 :: cmat(+6,+6) |
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| 82 | |
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| 83 | integer, parameter :: NSIZ = +6 |
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| 84 | real*8 , parameter :: ZERO = 1.d-14 |
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| 85 | |
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| 86 | head = +4 ; tail = npos - 3 |
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| 87 | |
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| 88 | if (npos.le.6) then |
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| 89 | !----- default to reduced order if insufficient points ! |
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| 90 | call p3e (npos,nvar,ndof, & |
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| 91 | & delx,fdat,bclo, & |
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| 92 | & bchi,edge,dfdx, & |
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| 93 | & opts,dmin) |
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| 94 | end if |
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| 95 | |
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| 96 | if (npos.le.6) return |
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| 97 | |
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| 98 | !------ impose value/slope B.C.'s about lower endpoint ! |
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| 99 | |
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| 100 | call pbc(npos,nvar,ndof,delx, & |
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| 101 | & fdat,bclo,edge,dfdx, & |
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| 102 | & -1 ,dmin) |
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| 103 | |
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| 104 | !------ impose value/slope B.C.'s about upper endpoint ! |
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| 105 | |
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| 106 | call pbc(npos,nvar,ndof,delx, & |
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| 107 | & fdat,bchi,edge,dfdx, & |
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| 108 | & +1 ,dmin) |
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| 109 | |
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| 110 | ! Reconstruct edge-centred 6th-order polynomials. Com- ! |
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| 111 | ! pute values/slopes at edges directly. Mid.-order ex- ! |
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| 112 | ! trapolation at endpoints. ! |
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| 113 | |
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| 114 | if (size(delx).eq.+1) then |
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| 115 | |
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| 116 | do ipos = head , tail |
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| 117 | |
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| 118 | !--------------- reconstruction: constant grid-spacing ! |
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| 119 | |
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| 120 | do ivar = 1, nvar |
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| 121 | |
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| 122 | edge(ivar,ipos) = & |
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| 123 | & + ( 1.d0 / 60.d0) * & |
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| 124 | & fdat(1,ivar,ipos-3) & |
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| 125 | & - ( 8.d0 / 60.d0) * & |
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| 126 | & fdat(1,ivar,ipos-2) & |
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| 127 | & + (37.d0 / 60.d0) * & |
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| 128 | & fdat(1,ivar,ipos-1) & |
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| 129 | & + (37.d0 / 60.d0) * & |
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| 130 | & fdat(1,ivar,ipos+0) & |
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| 131 | & - ( 8.d0 / 60.d0) * & |
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| 132 | & fdat(1,ivar,ipos+1) & |
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| 133 | & + ( 1.d0 / 60.d0) * & |
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| 134 | & fdat(1,ivar,ipos+2) |
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| 135 | |
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| 136 | dfdx(ivar,ipos) = & |
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| 137 | & - ( 1.d0 / 90.d0) * & |
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| 138 | & fdat(1,ivar,ipos-3) & |
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| 139 | & + ( 5.d0 / 36.d0) * & |
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| 140 | & fdat(1,ivar,ipos-2) & |
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| 141 | & - (49.d0 / 36.d0) * & |
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| 142 | & fdat(1,ivar,ipos-1) & |
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| 143 | & + (49.d0 / 36.d0) * & |
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| 144 | & fdat(1,ivar,ipos+0) & |
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| 145 | & - ( 5.d0 / 36.d0) * & |
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| 146 | & fdat(1,ivar,ipos+1) & |
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| 147 | & + ( 1.d0 / 90.d0) * & |
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| 148 | & fdat(1,ivar,ipos+2) |
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| 149 | |
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| 150 | dfdx(ivar,ipos) = & |
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| 151 | dfdx(ivar,ipos) / delx(+1) |
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| 152 | |
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| 153 | end do |
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| 154 | |
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| 155 | end do |
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| 156 | |
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| 157 | else |
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| 158 | |
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| 159 | fEPS = ZERO * dmin |
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| 160 | |
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| 161 | do ipos = head , tail |
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| 162 | |
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| 163 | !--------------- reconstruction: variable grid-spacing ! |
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| 164 | |
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| 165 | delh(-3) = & |
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| 166 | & max(delx(ipos-3),dmin) |
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| 167 | delh(-2) = & |
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| 168 | & max(delx(ipos-2),dmin) |
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| 169 | delh(-1) = & |
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| 170 | & max(delx(ipos-1),dmin) |
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| 171 | delh(+0) = & |
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| 172 | & max(delx(ipos+0),dmin) |
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| 173 | delh(+1) = & |
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| 174 | & max(delx(ipos+1),dmin) |
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| 175 | delh(+2) = & |
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| 176 | & max(delx(ipos+2),dmin) |
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| 177 | |
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| 178 | xhat = .5d0 * delh(-1) + & |
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| 179 | & .5d0 * delh(+0) |
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| 180 | |
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| 181 | xmap(-3) = -( delh(-3) & |
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| 182 | & + delh(-2) & |
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| 183 | & + delh(-1) ) / xhat |
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| 184 | xmap(-2) = -( delh(-2) & |
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| 185 | & + delh(-1) ) / xhat |
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| 186 | xmap(-1) = - delh(-1) / xhat |
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| 187 | xmap(+0) = + 0.d0 |
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| 188 | xmap(+1) = + delh(+0) / xhat |
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| 189 | xmap(+2) = +( delh(+0) & |
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| 190 | & + delh(+1) ) / xhat |
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| 191 | xmap(+3) = +( delh(+0) & |
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| 192 | & + delh(+1) & |
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| 193 | & + delh(+2) ) / xhat |
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| 194 | |
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| 195 | !--------------------------- calc. integral basis vec. ! |
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| 196 | |
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| 197 | do idof = -3, +3 |
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| 198 | |
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| 199 | ivec(1,idof) = & |
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| 200 | & xmap(idof) ** 1 / 1.0d+0 |
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| 201 | ivec(2,idof) = & |
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| 202 | & xmap(idof) ** 2 / 2.0d+0 |
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| 203 | ivec(3,idof) = & |
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| 204 | & xmap(idof) ** 3 / 3.0d+0 |
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| 205 | ivec(4,idof) = & |
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| 206 | & xmap(idof) ** 4 / 4.0d+0 |
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| 207 | ivec(5,idof) = & |
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| 208 | & xmap(idof) ** 5 / 5.0d+0 |
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| 209 | ivec(6,idof) = & |
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| 210 | & xmap(idof) ** 6 / 6.0d+0 |
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| 211 | |
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| 212 | end do |
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| 213 | |
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| 214 | !--------------------------- linear system: lhs matrix ! |
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| 215 | |
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| 216 | do idof = +1, +6 |
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| 217 | |
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| 218 | cmat(1,idof) = ivec(idof,-2) & |
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| 219 | & - ivec(idof,-3) |
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| 220 | cmat(2,idof) = ivec(idof,-1) & |
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| 221 | & - ivec(idof,-2) |
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| 222 | cmat(3,idof) = ivec(idof,+0) & |
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| 223 | & - ivec(idof,-1) |
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| 224 | cmat(4,idof) = ivec(idof,+1) & |
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| 225 | & - ivec(idof,+0) |
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| 226 | cmat(5,idof) = ivec(idof,+2) & |
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| 227 | & - ivec(idof,+1) |
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| 228 | cmat(6,idof) = ivec(idof,+3) & |
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| 229 | & - ivec(idof,+2) |
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| 230 | |
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| 231 | end do |
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| 232 | |
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| 233 | !--------------------------- linear system: rhs vector ! |
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| 234 | |
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| 235 | do ivar = +1, nvar |
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| 236 | |
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| 237 | fhat(+1,ivar) = & |
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| 238 | & delx(ipos-3) * & |
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| 239 | & fdat(+1,ivar,ipos-3) / xhat |
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| 240 | fhat(+2,ivar) = & |
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| 241 | & delx(ipos-2) * & |
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| 242 | & fdat(+1,ivar,ipos-2) / xhat |
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| 243 | fhat(+3,ivar) = & |
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| 244 | & delx(ipos-1) * & |
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| 245 | & fdat(+1,ivar,ipos-1) / xhat |
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| 246 | fhat(+4,ivar) = & |
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| 247 | & delx(ipos+0) * & |
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| 248 | & fdat(+1,ivar,ipos+0) / xhat |
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| 249 | fhat(+5,ivar) = & |
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| 250 | & delx(ipos+1) * & |
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| 251 | & fdat(+1,ivar,ipos+1) / xhat |
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| 252 | fhat(+6,ivar) = & |
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| 253 | & delx(ipos+2) * & |
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| 254 | & fdat(+1,ivar,ipos+2) / xhat |
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| 255 | |
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| 256 | end do |
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| 257 | |
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| 258 | !------------------------- factor/solve linear systems ! |
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| 259 | |
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| 260 | call slv_6x6(cmat,NSIZ,fhat, & |
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| 261 | & NSIZ,nvar,fEPS, & |
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| 262 | & okay) |
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| 263 | |
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| 264 | if (okay .eqv. .true.) then |
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| 265 | |
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| 266 | do ivar = +1, nvar |
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| 267 | |
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| 268 | edge(ivar,ipos) = fhat(1,ivar) |
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| 269 | |
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| 270 | dfdx(ivar,ipos) = fhat(2,ivar) & |
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| 271 | & / xhat |
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| 272 | |
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| 273 | end do |
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| 274 | |
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| 275 | else |
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| 276 | |
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| 277 | !------------------------- fallback if system singular ! |
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| 278 | |
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| 279 | # ifdef __PPR_WARNMAT__ |
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| 280 | |
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| 281 | write(*,*) & |
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| 282 | & "WARNING::P5E - matrix-is-singular!" |
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| 283 | |
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| 284 | # endif |
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| 285 | |
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| 286 | do ivar = +1, nvar |
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| 287 | |
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| 288 | edge(ivar,ipos) = & |
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| 289 | & fdat(1,ivar,ipos-1) * 0.5d+0 + & |
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| 290 | & fdat(1,ivar,ipos-0) * 0.5d+0 |
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| 291 | |
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| 292 | dfdx(ivar,ipos) = & |
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| 293 | & fdat(1,ivar,ipos-0) * 0.5d+0 - & |
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| 294 | & fdat(1,ivar,ipos-1) * 0.5d+0 |
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| 295 | |
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| 296 | dfdx(ivar,ipos) = & |
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| 297 | & dfdx(ivar,ipos) / xhat |
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| 298 | |
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| 299 | end do |
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| 300 | |
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| 301 | end if |
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| 302 | |
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| 303 | end do |
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| 304 | |
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| 305 | end if |
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| 306 | |
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| 307 | return |
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| 308 | |
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| 309 | end subroutine |
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| 310 | |
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| 311 | |
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| 312 | |
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