[13926] | 1 | |
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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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[14212] | 30 | ! PLM.h90: a 1d, slope-limited piecewise linear method. |
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[13926] | 31 | ! |
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| 32 | ! Darren Engwirda |
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| 33 | ! 08-Sep-2016 |
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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 | pure subroutine plm(npos,nvar,ndof,delx, & |
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| 39 | & fdat,fhat,dmin,ilim) |
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| 40 | |
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| 41 | ! |
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| 42 | ! NPOS no. edges over grid. |
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| 43 | ! NVAR no. state variables. |
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| 44 | ! NDOF no. degrees-of-freedom per grid-cell . |
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| 45 | ! DELX grid-cell spacing array. LENGTH(DELX) == +1 if |
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| 46 | ! spacing is uniform . |
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| 47 | ! FDAT grid-cell moments array. FDAT is an array with |
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| 48 | ! SIZE = NDOF-by-NVAR-by-NPOS-1 . |
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| 49 | ! FHAT grid-cell re-con. array. FHAT is an array with |
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| 50 | ! SIZE = MDOF-by-NVAR-by-NPOS-1 . |
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| 51 | ! DMIN min. grid-cell spacing thresh . |
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| 52 | ! ILIM cell slope-limiting selection . |
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| 53 | ! |
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| 54 | |
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| 55 | implicit none |
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| 56 | |
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| 57 | !------------------------------------------- arguments ! |
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| 58 | integer, intent( in) :: npos,nvar |
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| 59 | integer, intent( in) :: ndof,ilim |
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| 60 | real*8 , intent( in) :: dmin |
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| 61 | real*8 , intent( in) :: delx(:) |
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| 62 | real*8 , intent(out) :: fhat(:,:,:) |
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| 63 | real*8 , intent( in) :: fdat(:,:,:) |
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| 64 | |
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| 65 | if (size(delx).gt.+1) then |
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| 66 | |
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| 67 | !------------------------------- variable grid-spacing ! |
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| 68 | |
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| 69 | call plmv(npos,nvar,ndof,delx,& |
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| 70 | & fdat,fhat,& |
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| 71 | & dmin,ilim ) |
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| 72 | |
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| 73 | else |
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| 74 | |
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| 75 | !------------------------------- constant grid-spacing ! |
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| 76 | |
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| 77 | call plmc(npos,nvar,ndof,delx,& |
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| 78 | & fdat,fhat,& |
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| 79 | & dmin,ilim ) |
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| 80 | |
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| 81 | end if |
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| 82 | |
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| 83 | return |
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| 84 | |
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| 85 | end subroutine |
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| 86 | |
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| 87 | !------------------------- assemble PLM reconstruction ! |
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| 88 | |
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| 89 | pure subroutine plmv(npos,nvar,ndof,delx, & |
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| 90 | & fdat,fhat,dmin,ilim) |
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| 91 | |
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| 92 | ! |
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| 93 | ! *this is the variable grid-spacing variant . |
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| 94 | ! |
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| 95 | ! NPOS no. edges over grid. |
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| 96 | ! NVAR no. state variables. |
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| 97 | ! NDOF no. degrees-of-freedom per grid-cell . |
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| 98 | ! DELX grid-cell spacing array. LENGTH(DELX) == +1 if |
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| 99 | ! spacing is uniform . |
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| 100 | ! FDAT grid-cell moments array. FDAT is an array with |
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| 101 | ! SIZE = NDOF-by-NVAR-by-NPOS-1 . |
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| 102 | ! FHAT grid-cell re-con. array. FHAT is an array with |
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| 103 | ! SIZE = MDOF-by-NVAR-by-NPOS-1 . |
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| 104 | ! DMIN min. grid-cell spacing thresh . |
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| 105 | ! ILIM cell slope-limiting selection . |
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| 106 | ! |
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| 107 | |
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| 108 | implicit none |
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| 109 | |
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| 110 | !------------------------------------------- arguments ! |
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| 111 | integer, intent( in) :: npos,nvar |
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| 112 | integer, intent( in) :: ndof,ilim |
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| 113 | real*8 , intent( in) :: dmin |
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| 114 | real*8 , intent( in) :: delx(:) |
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| 115 | real*8 , intent(out) :: fhat(:,:,:) |
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| 116 | real*8 , intent( in) :: fdat(:,:,:) |
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| 117 | |
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| 118 | !------------------------------------------- variables ! |
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| 119 | integer :: ipos,ivar,head,tail |
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| 120 | real*8 :: dfds(-1:+1) |
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| 121 | |
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| 122 | head = +1; tail = npos - 1 |
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| 123 | |
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| 124 | if (npos.eq.2) then |
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| 125 | !----------------------- reduce order if small stencil ! |
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| 126 | do ivar = +1, nvar |
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| 127 | fhat(1,ivar,1) = & |
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| 128 | & fdat(1,ivar,1) |
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| 129 | fhat(2,ivar,1) = 0.d+0 |
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| 130 | end do |
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| 131 | end if |
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| 132 | |
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| 133 | if (npos.le.2) return |
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| 134 | |
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| 135 | !-------------------------------------- lower-endpoint ! |
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| 136 | |
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| 137 | do ivar = +1 , nvar-0 |
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| 138 | |
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| 139 | call plsv( & |
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| 140 | & fdat(1,ivar,head+0) , & |
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| 141 | & delx(head+0), & |
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| 142 | & fdat(1,ivar,head+0) , & |
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| 143 | & delx(head+0), & |
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| 144 | & fdat(1,ivar,head+1) , & |
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| 145 | & delx(head+1), dfds) |
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| 146 | |
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| 147 | fhat(1,ivar,head) = & |
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| 148 | & fdat(1,ivar,head) |
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| 149 | fhat(2,ivar,head) = dfds(0) |
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| 150 | |
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| 151 | end do |
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| 152 | |
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| 153 | !-------------------------------------- upper-endpoint ! |
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| 154 | |
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| 155 | do ivar = +1 , nvar-0 |
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| 156 | |
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| 157 | call plsv( & |
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| 158 | & fdat(1,ivar,tail-1) , & |
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| 159 | & delx(tail-1), & |
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| 160 | & fdat(1,ivar,tail+0) , & |
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| 161 | & delx(tail+0), & |
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| 162 | & fdat(1,ivar,tail+0) , & |
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| 163 | & delx(tail+0), dfds) |
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| 164 | |
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| 165 | fhat(1,ivar,tail) = & |
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| 166 | & fdat(1,ivar,tail) |
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| 167 | fhat(2,ivar,tail) = dfds(0) |
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| 168 | |
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| 169 | end do |
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| 170 | |
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| 171 | !-------------------------------------- interior cells ! |
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| 172 | |
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| 173 | do ipos = +2 , npos-2 |
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| 174 | do ivar = +1 , nvar-0 |
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| 175 | |
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| 176 | call plsv( & |
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| 177 | & fdat(1,ivar,ipos-1) , & |
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| 178 | & delx(ipos-1), & |
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| 179 | & fdat(1,ivar,ipos+0) , & |
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| 180 | & delx(ipos+0), & |
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| 181 | & fdat(1,ivar,ipos+1) , & |
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| 182 | & delx(ipos+1), dfds) |
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| 183 | |
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| 184 | fhat(1,ivar,ipos) = & |
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| 185 | & fdat(1,ivar,ipos) |
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| 186 | fhat(2,ivar,ipos) = dfds(0) |
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| 187 | |
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| 188 | end do |
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| 189 | end do |
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| 190 | |
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| 191 | return |
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| 192 | |
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| 193 | end subroutine |
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| 194 | |
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| 195 | !------------------------- assemble PLM reconstruction ! |
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| 196 | |
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| 197 | pure subroutine plmc(npos,nvar,ndof,delx, & |
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| 198 | & fdat,fhat,dmin,ilim) |
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| 199 | |
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| 200 | ! |
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| 201 | ! *this is the constant grid-spacing variant . |
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| 202 | ! |
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| 203 | ! NPOS no. edges over grid. |
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| 204 | ! NVAR no. state variables. |
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| 205 | ! NDOF no. degrees-of-freedom per grid-cell . |
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| 206 | ! DELX grid-cell spacing array. LENGTH(DELX) == +1 if |
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| 207 | ! spacing is uniform . |
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| 208 | ! FDAT grid-cell moments array. FDAT is an array with |
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| 209 | ! SIZE = NDOF-by-NVAR-by-NPOS-1 . |
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| 210 | ! FHAT grid-cell re-con. array. FHAT is an array with |
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| 211 | ! SIZE = MDOF-by-NVAR-by-NPOS-1 . |
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| 212 | ! DMIN min. grid-cell spacing thresh . |
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| 213 | ! ILIM cell slope-limiting selection . |
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| 214 | ! |
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| 215 | |
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| 216 | implicit none |
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| 217 | |
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| 218 | !------------------------------------------- arguments ! |
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| 219 | integer, intent( in) :: npos,nvar |
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| 220 | integer, intent( in) :: ndof,ilim |
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| 221 | real*8 , intent( in) :: dmin |
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| 222 | real*8 , intent( in) :: delx(1) |
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| 223 | real*8 , intent(out) :: fhat(:,:,:) |
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| 224 | real*8 , intent( in) :: fdat(:,:,:) |
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| 225 | |
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| 226 | !------------------------------------------- variables ! |
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| 227 | integer :: ipos,ivar,head,tail |
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| 228 | real*8 :: dfds(-1:+1) |
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| 229 | |
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| 230 | head = +1; tail = npos - 1 |
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| 231 | |
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| 232 | if (npos.eq.2) then |
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| 233 | !----------------------- reduce order if small stencil ! |
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| 234 | do ivar = +1, nvar |
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| 235 | fhat(1,ivar,1) = & |
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| 236 | & fdat(1,ivar,1) |
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| 237 | fhat(2,ivar,1) = 0.d+0 |
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| 238 | end do |
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| 239 | end if |
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| 240 | |
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| 241 | if (npos.le.2) return |
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| 242 | |
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| 243 | !-------------------------------------- lower-endpoint ! |
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| 244 | |
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| 245 | do ivar = +1 , nvar-0 |
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| 246 | |
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| 247 | call plsc( & |
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| 248 | & fdat(1,ivar,head+0) , & |
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| 249 | & fdat(1,ivar,head+0) , & |
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| 250 | & fdat(1,ivar,head+1) , & |
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| 251 | & dfds) |
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| 252 | |
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| 253 | fhat(1,ivar,head) = & |
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| 254 | & fdat(1,ivar,head) |
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| 255 | fhat(2,ivar,head) = dfds(0) |
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| 256 | |
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| 257 | end do |
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| 258 | |
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| 259 | !-------------------------------------- upper-endpoint ! |
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| 260 | |
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| 261 | do ivar = +1 , nvar-0 |
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| 262 | |
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| 263 | call plsc( & |
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| 264 | & fdat(1,ivar,tail-1) , & |
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| 265 | & fdat(1,ivar,tail+0) , & |
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| 266 | & fdat(1,ivar,tail+0) , & |
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| 267 | & dfds) |
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| 268 | |
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| 269 | fhat(1,ivar,tail) = & |
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| 270 | & fdat(1,ivar,tail) |
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| 271 | fhat(2,ivar,tail) = dfds(0) |
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| 272 | |
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| 273 | end do |
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| 274 | |
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| 275 | !-------------------------------------- interior cells ! |
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| 276 | |
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| 277 | do ipos = +2 , npos-2 |
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| 278 | do ivar = +1 , nvar-0 |
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| 279 | |
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| 280 | call plsc( & |
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| 281 | & fdat(1,ivar,ipos-1) , & |
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| 282 | & fdat(1,ivar,ipos+0) , & |
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| 283 | & fdat(1,ivar,ipos+1) , & |
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| 284 | & dfds) |
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| 285 | |
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| 286 | fhat(1,ivar,ipos) = & |
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| 287 | & fdat(1,ivar,ipos) |
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| 288 | fhat(2,ivar,ipos) = dfds(0) |
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| 289 | |
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| 290 | end do |
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| 291 | end do |
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| 292 | |
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| 293 | return |
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| 294 | |
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| 295 | end subroutine |
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| 296 | |
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| 297 | !------------------------------- assemble PLM "slopes" ! |
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| 298 | |
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| 299 | pure subroutine plsv(ffll,hhll,ff00,hh00,& |
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| 300 | & ffrr,hhrr,dfds) |
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| 301 | |
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| 302 | ! |
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| 303 | ! *this is the variable grid-spacing variant . |
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| 304 | ! |
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| 305 | ! FFLL left -biased grid-cell mean. |
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| 306 | ! HHLL left -biased grid-cell spac. |
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| 307 | ! FF00 centred grid-cell mean. |
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| 308 | ! HH00 centred grid-cell spac. |
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| 309 | ! FFRR right-biased grid-cell mean. |
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| 310 | ! HHRR right-biased grid-cell spac. |
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| 311 | ! DFDS piecewise linear gradients in local co-ord.'s. |
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| 312 | ! DFDS(+0) is a centred, slope-limited estimate, |
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| 313 | ! DFDS(-1), DFDS(+1) are left- and right-biased |
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| 314 | ! estimates (unlimited). |
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| 315 | ! |
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| 316 | |
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| 317 | implicit none |
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| 318 | |
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| 319 | !------------------------------------------- arguments ! |
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| 320 | real*8 , intent( in) :: ffll,ff00,ffrr |
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| 321 | real*8 , intent( in) :: hhll,hh00,hhrr |
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| 322 | real*8 , intent(out) :: dfds(-1:+1) |
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| 323 | |
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| 324 | !------------------------------------------- variables ! |
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| 325 | real*8 :: fell,ferr,scal |
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| 326 | |
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| 327 | real*8 , parameter :: ZERO = 1.d-14 |
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| 328 | |
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| 329 | !---------------------------- 2nd-order approximations ! |
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| 330 | |
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| 331 | dfds(-1) = ff00-ffll |
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| 332 | dfds(+1) = ffrr-ff00 |
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| 333 | |
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| 334 | if (dfds(-1) * & |
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| 335 | & dfds(+1) .gt. 0.0d+0) then |
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| 336 | |
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| 337 | !---------------------------- calc. ll//rr edge values ! |
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| 338 | |
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| 339 | fell = (hh00*ffll+hhll*ff00) & |
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| 340 | & / (hhll+hh00) |
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| 341 | ferr = (hhrr*ff00+hh00*ffrr) & |
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| 342 | & / (hh00+hhrr) |
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| 343 | |
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| 344 | !---------------------------- calc. centred derivative ! |
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| 345 | |
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| 346 | dfds(+0) = & |
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| 347 | & 0.5d+0 * (ferr - fell) |
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| 348 | |
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| 349 | !---------------------------- monotonic slope-limiting ! |
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| 350 | |
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| 351 | scal = min(abs(dfds(-1)), & |
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| 352 | & abs(dfds(+1))) & |
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| 353 | & / max(abs(dfds(+0)), & |
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| 354 | ZERO) |
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| 355 | scal = min(scal,+1.0d+0) |
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| 356 | |
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| 357 | dfds(+0) = scal * dfds(+0) |
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| 358 | |
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| 359 | else |
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| 360 | |
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| 361 | !---------------------------- flatten if local extrema ! |
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| 362 | |
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| 363 | dfds(+0) = +0.0d+0 |
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| 364 | |
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| 365 | end if |
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| 366 | |
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| 367 | !---------------------------- scale onto local co-ord. ! |
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| 368 | |
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| 369 | dfds(-1) = dfds(-1) & |
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| 370 | & / (hhll + hh00) * hh00 |
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| 371 | dfds(+1) = dfds(+1) & |
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| 372 | & / (hh00 + hhrr) * hh00 |
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| 373 | |
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| 374 | return |
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| 375 | |
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| 376 | end subroutine |
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| 377 | |
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| 378 | !------------------------------- assemble PLM "slopes" ! |
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| 379 | |
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| 380 | pure subroutine plsc(ffll,ff00,ffrr,dfds) |
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| 381 | |
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| 382 | ! |
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| 383 | ! *this is the constant grid-spacing variant . |
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| 384 | ! |
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| 385 | ! FFLL left -biased grid-cell mean. |
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| 386 | ! FF00 centred grid-cell mean. |
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| 387 | ! FFRR right-biased grid-cell mean. |
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| 388 | ! DFDS piecewise linear gradients in local co-ord.'s. |
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| 389 | ! DFDS(+0) is a centred, slope-limited estimate, |
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| 390 | ! DFDS(-1), DFDS(+1) are left- and right-biased |
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| 391 | ! estimates (unlimited). |
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| 392 | ! |
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| 393 | |
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| 394 | implicit none |
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| 395 | |
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| 396 | !------------------------------------------- arguments ! |
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| 397 | real*8 , intent( in) :: ffll,ff00,ffrr |
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| 398 | real*8 , intent(out) :: dfds(-1:+1) |
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| 399 | |
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| 400 | !------------------------------------------- variables ! |
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| 401 | real*8 :: fell,ferr,scal |
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| 402 | |
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| 403 | real*8 , parameter :: ZERO = 1.d-14 |
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| 404 | |
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| 405 | !---------------------------- 2nd-order approximations ! |
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| 406 | |
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| 407 | dfds(-1) = ff00-ffll |
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| 408 | dfds(+1) = ffrr-ff00 |
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| 409 | |
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| 410 | if (dfds(-1) * & |
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| 411 | & dfds(+1) .gt. 0.0d+0) then |
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| 412 | |
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| 413 | !---------------------------- calc. ll//rr edge values ! |
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| 414 | |
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| 415 | fell = (ffll+ff00) * .5d+0 |
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| 416 | ferr = (ff00+ffrr) * .5d+0 |
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| 417 | |
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| 418 | !---------------------------- calc. centred derivative ! |
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| 419 | |
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| 420 | dfds(+0) = & |
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| 421 | & 0.5d+0 * (ferr - fell) |
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| 422 | |
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| 423 | !---------------------------- monotonic slope-limiting ! |
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| 424 | |
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| 425 | scal = min(abs(dfds(-1)), & |
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| 426 | & abs(dfds(+1))) & |
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| 427 | & / max(abs(dfds(+0)), & |
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| 428 | ZERO) |
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| 429 | scal = min(scal,+1.0d+0) |
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| 430 | |
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| 431 | dfds(+0) = scal * dfds(+0) |
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| 432 | |
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| 433 | else |
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| 434 | |
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| 435 | !---------------------------- flatten if local extrema ! |
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| 436 | |
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| 437 | dfds(+0) = +0.0d+0 |
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| 438 | |
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| 439 | end if |
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| 440 | |
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| 441 | !---------------------------- scale onto local co-ord. ! |
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| 442 | |
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| 443 | dfds(-1) = + 0.5d+0 * dfds(-1) |
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| 444 | dfds(+1) = + 0.5d+0 * dfds(+1) |
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| 445 | |
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| 446 | return |
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| 447 | |
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| 448 | end subroutine |
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| 449 | |
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| 450 | |
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| 451 | |
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