Page 1                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



     NNNNAAAAMMMMEEEE
          ddddoooo____ppppppppmmmmllllrrrr0000____3333ddddcccc____ggggaaaammmmmmmmaaaa - Given 1d strips of zone averages of
          density, pressure, and velocity defined on a uniform grid,
          new zone averages of these quantities are returned by
          advancing the Euler equations forward one timestep in the X
          direction. The full2-dimensional update requires directional
          splitting by the calling program.



     FFFFOOOORRRRTTTTRRRRAAAANNNN SSSSPPPPEEEECCCCIIIIFFFFIIIICCCCAAAATTTTIIIIOOOONNNN
          ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee ddddoooo____ppppppppmmmmllllrrrr0000____3333ddddcccc____ggggaaaammmmmmmmaaaa(((( XXXXLLLL,,,, RRRRHHHHOOOOZZZZBBBBBBBB,,,, RRRRHHHHOOOO,,,, RRRRHHHHOOOOZZZZTTTTTTTT,,,,
          RRRRHHHHOOOOZZZZMMMMMMMM,,,, RRRRHHHHOOOOZZZZPPPPPPPP,,,, PPPPZZZZBBBBBBBB,,,, PPPP,,,, PPPPZZZZTTTTTTTT,,,, PPPPZZZZMMMMMMMM,,,, PPPPZZZZPPPPPPPP,,,, UUUU1111ZZZZBBBBBBBB,,,, UUUU1111,,,,
          UUUU1111ZZZZTTTTTTTT,,,, UUUU1111ZZZZMMMMMMMM,,,, UUUU1111ZZZZPPPPPPPP,,,, UUUU2222ZZZZBBBBBBBB,,,, UUUU2222ZZZZBBBB,,,, UUUU2222,,,, UUUU2222ZZZZTTTT,,,, UUUU2222ZZZZTTTTTTTT,,,,
          UUUU2222ZZZZMMMMMMMM,,,, UUUU2222ZZZZPPPPPPPP,,,, UUUU3333ZZZZMMMMMMMM,,,, UUUU3333ZZZZMMMM,,,, UUUU3333,,,, UUUU3333ZZZZPPPP,,,, UUUU3333ZZZZPPPPPPPP,,,, UUUU3333ZZZZBBBBBBBB,,,,
          UUUU3333ZZZZTTTTTTTT,,,, RRRRHHHHOOOONNNNUUUU,,,, PPPPNNNNUUUU,,,, UUUU1111NNNNUUUU,,,, UUUU2222NNNNUUUU,,,, UUUU3333NNNNUUUU,,,, EEEEOOOOSSSSGGGGAAAAMMMM,,,, DDDDTTTT,,,, SSSSMMMMLLLLRRRRHHHHOOOO,,,,
          SSSSMMMMAAAALLLLLLLLPPPP,,,, SSSSMMMMAAAALLLLLLLLUUUU,,,, SSSSMMMMAAAALLLLLLLLEEEE,,,, CCCCOOOOUUUURRRRMMMMXXXX,,,, NNNNZZZZOOOONNNNEEEESSSS,,,, NNNNBBBBDDDDYYYY ))))
          rrrreeeeaaaallll****4444    XXXXLLLL((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY++++1111))))
          rrrreeeeaaaallll****4444 RRRRHHHHOOOOZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444   RRRRHHHHOOOO((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 RRRRHHHHOOOOZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 RRRRHHHHOOOOZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 RRRRHHHHOOOOZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  PPPPZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444     PPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  PPPPZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  PPPPZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  PPPPZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU1111ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444    UUUU1111((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU1111ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU1111ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU1111ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU2222ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU2222ZZZZBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444    UUUU2222((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU2222ZZZZTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU2222ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU2222ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU2222ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU3333ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444    UUUU3333((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU3333ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU3333ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU3333ZZZZMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU3333ZZZZPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 UUUU3333ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444 RRRRHHHHOOOONNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444   PPPPNNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU1111NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU2222NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****4444  UUUU3333NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))



     Page 1                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          rrrreeeeaaaallll****4444 EEEEOOOOSSSSGGGGAAAAMMMM,,,, DDDDTTTT,,,, SSSSMMMMLLLLRRRRHHHHOOOO,,,, SSSSMMMMAAAALLLLLLLLPPPP.... SSSSMMMMAAAALLLLLLLLUUUU,,,, SSSSMMMMAAAALLLLLLLLEEEE,,,,
          rrrreeeeaaaallll****4444 CCCCOOOOUUUURRRRMMMMXXXX
          iiiinnnntttteeeeggggeeeerrrr****4444 NNNNZZZZOOOONNNNEEEESSSS,,,, NNNNBBBBDDDDYYYY



          ssssuuuubbbbrrrroooouuuuttttiiiinnnneeee 3333____ddddoooo____ppppppppmmmmllllrrrr0000____3333ddddcccc____ggggaaaammmmmmmmaaaa(((( XXXXLLLL,,,, RRRRHHHHOOOOZZZZBBBBBBBB,,,, RRRRHHHHOOOO,,,, RRRRHHHHOOOOZZZZTTTTTTTT,,,,
          RRRRHHHHOOOOZZZZMMMMMMMM,,,, RRRRHHHHOOOOZZZZPPPPPPPP,,,, PPPPZZZZBBBBBBBB,,,, PPPP,,,, PPPPZZZZTTTTTTTT,,,, PPPPZZZZMMMMMMMM,,,, PPPPZZZZPPPPPPPP,,,, UUUU1111ZZZZBBBBBBBB,,,, UUUU1111,,,,
          UUUU1111ZZZZTTTTTTTT,,,, UUUU1111ZZZZMMMMMMMM,,,, UUUU1111ZZZZPPPPPPPP,,,, UUUU2222ZZZZBBBBBBBB,,,, UUUU2222ZZZZBBBB,,,, UUUU2222,,,, UUUU2222ZZZZTTTT,,,, UUUU2222ZZZZTTTTTTTT,,,,
          UUUU2222ZZZZMMMMMMMM,,,, UUUU2222ZZZZPPPPPPPP,,,, UUUU3333ZZZZMMMMMMMM,,,, UUUU3333ZZZZMMMM,,,, UUUU3333,,,, UUUU3333ZZZZPPPP,,,, UUUU3333ZZZZPPPPPPPP,,,, UUUU3333ZZZZBBBBBBBB,,,,
          UUUU3333ZZZZTTTTTTTT,,,, RRRRHHHHOOOONNNNUUUU,,,, PPPPNNNNUUUU,,,, UUUU1111NNNNUUUU,,,, UUUU2222NNNNUUUU,,,, UUUU3333NNNNUUUU,,,, EEEEOOOOSSSSGGGGAAAAMMMM,,,, DDDDTTTT,,,, SSSSMMMMLLLLRRRRHHHHOOOO,,,,
          SSSSMMMMAAAALLLLLLLLPPPP,,,, SSSSMMMMAAAALLLLLLLLUUUU,,,, SSSSMMMMAAAALLLLLLLLEEEE,,,, CCCCOOOOUUUURRRRMMMMXXXX,,,, NNNNZZZZOOOONNNNEEEESSSS,,,, NNNNBBBBDDDDYYYY ))))
          rrrreeeeaaaallll****8888    XXXXLLLL((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY++++1111))))
          rrrreeeeaaaallll****8888 RRRRHHHHOOOOZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888   RRRRHHHHOOOO((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 RRRRHHHHOOOOZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 RRRRHHHHOOOOZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 RRRRHHHHOOOOZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  PPPPZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888     PPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  PPPPZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  PPPPZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  PPPPZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU1111ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888    UUUU1111((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU1111ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU1111ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU1111ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU2222ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU2222ZZZZBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888    UUUU2222((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU2222ZZZZTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU2222ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU2222ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU2222ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU3333ZZZZBBBBBBBB((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888    UUUU3333((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU3333ZZZZTTTTTTTT((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU3333ZZZZMMMMMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU3333ZZZZMMMM((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU3333ZZZZPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 UUUU3333ZZZZPPPPPPPP((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 RRRRHHHHOOOONNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888   PPPPNNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU1111NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU2222NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888  UUUU3333NNNNUUUU((((1111----NNNNBBBBDDDDYYYY::::NNNNZZZZOOOONNNNEEEESSSS++++NNNNBBBBDDDDYYYY))))
          rrrreeeeaaaallll****8888 EEEEOOOOSSSSGGGGAAAAMMMM,,,, DDDDTTTT,,,, SSSSMMMMLLLLRRRRHHHHOOOO,,,, SSSSMMMMAAAALLLLLLLLPPPP.... SSSSMMMMAAAALLLLLLLLUUUU,,,, SSSSMMMMAAAALLLLLLLLEEEE,,,,
          rrrreeeeaaaallll****8888 CCCCOOOOUUUURRRRMMMMXXXX
          iiiinnnntttteeeeggggeeeerrrr****4444 NNNNZZZZOOOONNNNEEEESSSS,,,, NNNNBBBBDDDDYYYY





     Page 2                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          These two routines are functionally the same.  They differ
          only in the type declarations of their (real) arguments
          (that is, single or double precision.)




     AAAARRRRGGGGUUUUMMMMEEEENNNNTTTTSSSS
          XXXXLLLL [input]
                    array of the locations of the lefthand zone
                    interfaces. This defines the grid.  Values on the
                    range [1-NBDY, NZONES+NBDY+1] are required


          RRRRHHHHOOOOZZZZBBBBBBBB [input]
                    array of zone averages of density, two rows below
                    and parallel to the row to be updated.  Values on
                    the range [1-NBDY, NZONES+NBDY] are required


          RRRRHHHHOOOO [input]
                    array of zone averages of density.  Values on the
                    range [1-NBDY, NZONES+NBDY] are required


          RRRRHHHHOOOOZZZZTTTTTTTT [input]
                    array of zone averages of density, two rows above
                    and parallel to the row to be updated.  Values on
                    the range [1-NBDY, NZONES+NBDY] are required


          RRRRHHHHOOOOZZZZMMMMMMMM [input]
                    array of zone averages of density, two planes
                    behind and parallel to the row to be updated.
                    Values on the range [1-NBDY, NZONES+NBDY] are
                    required


          RRRRHHHHOOOOZZZZPPPPPPPP [input]
                    array of zone averages of density, two planes
                    before and parallel to the row to be updated.
                    Values on the range [1-NBDY, NZONES+NBDY] are
                    required


          PPPPZZZZBBBBBBBB [input]
                    array of zone averages of pressure, two rows below
                    and parallel to the row to be updated.  Values on
                    the range [1-NBDY, NZONES+NBDY] are required


          PPPP [input] array of zone averages of pressure.  Values on the



     Page 3                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



                    range [1-NBDY, NZONES+NBDY] are required


          PPPPZZZZTTTTTTTT [input]
                    array of zone averages of pressure, two rows above
                    and parallel to the row to be updated.  Values on
                    the range [1-NBDY, NZONES+NBDY] are required


          PPPPZZZZMMMMMMMM [input]
                    array of zone averages of pressure, two planes
                    behind and parallel to the row to be updated.
                    Values on the range [1-NBDY, NZONES+NBDY] are
                    required


          PPPPZZZZPPPPPPPP [input]
                    array of zone averages of pressure, two planes
                    behind and parallel to the row to be updated.
                    Values on the range [1-NBDY, NZONES+NBDY] are
                    required








          UUUU1111ZZZZBBBBBBBB [input]
                    array of zone averages of longitudinal (u1)
                    velocity, two rows below and parallel to the row
                    to be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU1111ZZZZBBBB [input]
                    array of zone averages of longitudinal (u1)
                    velocity, one row below and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU1111 [input]
                    array of zone averages of longitudinal (u1)
                    velocity.  That is, the velocity, a vector
                    quantity, is resolved into three orthogonal
                    components, one parallel to the direction of the
                    pass (longitudinal-u1) and two perpendicular to
                    the direction of the pass, and to each other
                    (transverse velocities u2, u3.)  Values on the
                    range [1-NBDY, NZONES+NBDY] are required



     Page 4                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          UUUU1111ZZZZTTTT [input]
                    array of zone averages of longitudinal (u1)
                    velocity, one row above and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU1111ZZZZTTTTTTTT [input]
                    array of zone averages of longitudinal (u1)
                    velocity, two rows above and parallel to the row
                    to be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU1111ZZZZMMMMMMMM [input]
                    array of zone averages of longitudinal (u1)
                    velocity, two planes behind and parallel to the
                    row to be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU1111ZZZZPPPPPPPP [input]
                    array of zone averages of longitudinal (u1)
                    velocity, two planes before and parallel to the
                    row to be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required













          UUUU2222ZZZZBBBBBBBB [input]
                    array of zone averages of transverse velocity
                    (u2), two rows below and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU2222ZZZZBBBB [input]
                    array of zone averages of transverse velocity
                    (u2), one row below and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required




     Page 5                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          UUUU2222 [input]
                    array of zone averages of transverse velocity
                    (u2).  That is, the velocity, a vector quantity,
                    is resolved into two orthogonal components, one
                    parallel to the direction of the pass
                    (longitudinal) and one perpendicular to the
                    direction of the pass (transverse.)  Values on the
                    range [1-NBDY, NZONES+NBDY] are required


          UUUU2222ZZZZTTTT [input]
                    array of zone averages of transverse velocity
                    (u2), one row above and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU2222ZZZZTTTTTTTT [input]
                    array of zone averages of transverse velocity
                    (u2), two rows above and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU2222ZZZZMMMMMMMM [input]
                    array of zone averages of transverse velocity
                    (u2), two planes behind and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU2222ZZZZPPPPPPPP [input]
                    array of zone averages of transverse velocity
                    (u2), two planes before and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required



















     Page 6                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          UUUU3333ZZZZBBBBBBBB [input]
                    array of zone averages of transverse velocity
                    (u3), two rows below and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU3333 [input]
                    array of zone averages of transverse velocity
                    (u3).  That is, the velocity, a vector quantity,
                    is resolved into two orthogonal components, one
                    parallel to the direction of the pass
                    (longitudinal) and one perpendicular to the
                    direction of the pass (transverse.)  Values on the
                    range [1-NBDY, NZONES+NBDY] are required


          UUUU3333ZZZZTTTTTTTT [input]
                    array of zone averages of transverse velocity
                    (u3), two rows above and parallel to the row to be
                    updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU3333ZZZZMMMMMMMM [input]
                    array of zone averages of transverse velocity
                    (u3), two planes behind and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU3333ZZZZMMMM [input]
                    array of zone averages of transverse velocity
                    (u3), one plane behind and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU3333ZZZZPPPP [input]
                    array of zone averages of transverse velocity
                    (u3), one plane before and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required


          UUUU3333ZZZZPPPPPPPP [input]
                    array of zone averages of transverse velocity
                    (u3), two planes before and parallel to the row to
                    be updated.  Values on the range [1-NBDY,
                    NZONES+NBDY] are required





     Page 7                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          RRRRHHHHOOOONNNNUUUU [output]
                    array of updated zone averages of density.  Values
                    on the range [1, NZONES] are computed


          PPPPNNNNUUUU [output]
                    array of updated zone averages of pressure.
                    Values on the range [1, NZONES] are computed


          UUUU1111NNNNUUUU [output]
                    array of updated zone averages of longitudinal
                    (u1) velocity.  Values on the range [1, NZONES]
                    are computed


          UUUU2222NNNNUUUU [output]
                    array of updated zone averages of transverse (u2)
                    velocity.  Values on the range [1, NZONES] are
                    computed


          UUUU3333NNNNUUUU [output]
                    array of updated zone averages of transverse (u3)
                    velocity.  Values on the range [1, NZONES] are
                    computed



          EEEEOOOOSSSSGGGGAAAAMMMM [input]
                    the ratio of specific heats.  A gamma-law equation
                    of state is used in which p(i)=(gamma-
                    1)*rho(i)*ei(i), with ei being the specific
                    internal energy.


          DDDDTTTT [input]
                    the timestep.


          SSSSMMMMLLLLRRRRHHHHOOOO [input]
                    This indicates what should be considered a trivial
                    values of density.  It is used to reduce the
                    effects of roundoff errors.  Reasonable values are
                    1.0e-06 for single precision and 1.0e-08 for
                    double precision.


          SSSSMMMMAAAALLLLLLLLPPPP [input]
                    This indicates what should be considered a trivial
                    values of pressure.  It is used to reduce the
                    effects of roundoff errors.  Reasonable values are



     Page 8                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



                    1.0e-06 for single precision and 1.0e-08 for
                    double precision.


          SSSSMMMMAAAALLLLLLLLUUUU [input]
                    This indicates what should be considered a trivial
                    values of velocity.  It is used to reduce the
                    effects of roundoff errors.  Reasonable values are
                    1.0e-06 for single precision and 1.0e-08 for
                    double precision.


          SSSSMMMMAAAALLLLLLLLEEEE [input]
                    This indicates what should be considered a trivial
                    values of energy.  It is used to reduce the
                    effects of roundoff errors.  Reasonable values are
                    1.0e-06 for single precision and 1.0e-08 for
                    double precision.


          CCCCOOOOUUUURRRRMMMMXXXX [input]
                    The maximum Courant number encountered during this
                    call. The original value of COURMX is lost during
                    the computation. The Courant number is a stability
                    condition and should be less than one.


          NNNNZZZZOOOONNNNEEEESSSS [input]
                    the array dimension, or number of zones to
                    process, not counting boundary or ``fake'' zones.


          NNNNBBBBDDDDYYYY [input]
                    the number of boundary of ``fake'' zones which
                    have been set at each end of the 1d arrays to
                    implement the users choice of boundary conditions.
                    This must be atleast 5.



     DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN
          These routines are examples of one-dimensional hydrodynamics
          subroutines written a Lagrange plus remap style using
          elements from the PPMLIB library.  This routine is suitable
          for three-dimensional problems when combined with the
          technique of Directional splitting.  In the Lagrangian step,
          piecewise Parabolic interpolations are constructed for
          pressure and longitudinal velocity.  Where the input data
          distribution is not well represented by parabolae or poorly
          resolved, (that is, the data are ``unsmooth''), monotonicity
          constraints are applied.




     Page 9                                          (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          Next, a Riemann solver is used to resolve these resultant
          discontinuities into time-averaged fluxes at the zone
          interfaces.  These fluxes are differenced conservatively to
          provide new zone-averaged values of density, pressure (or
          energy), and velocity.  The transverse velocity is unchanged
          at this point.



          At the beginning of the remap step, new piecewise Parabolic
          interpolations are constructed for pressure and longitudinal
          velocity, using the updated zone-averages from the
          Lagrangian step.  Where the data are not well represented by
          parabolae or poorly resolved, (that is, the data are
          ``unsmooth''), monotonicity constraints are applied.
          Piecewise Parabolic interpolations are constructed for the
          density.  Contact discontinuity detection is performed and
          where it is determined likely that a contact discontinuity
          exists, the density interpolations are modified to better
          represent discontinuities, rather than a smooth
          distribution.  Monotonicity constrains are applied where the
          density data are deemed unsmooth.


          Next, regions of overlap between the Eulerian grid (the
          original grid) and the Lagrangian grid are found.  The
          piecewise parabolic interpolations found above are
          integrated over these overlap regions to produce advection
          fluxes.  These fluxes are differenced conservatively to
          provide new zone-averaged values of density, pressure (or
          energy), and longitudinal velocity.



          The transverse velocities are interpolated with monotonicity
          constraints.  These interpolations are integrated over the
          overlap regions and combined with the volume or mass fluxes
          found above, as appropriate, in construct the advected
          transverse momenta fluxes.  These fluxes are then
          differenced conservatively producing new transverse velocity
          zone-averages.



          Finally, a diffusion step is applied. Shock detection is
          performed, so that the additional diffusion is applied only
          in those zone which require it.  This includes strong shocks
          and certain pathological case such as a shock moving slowly
          through the numerical grid.

          In addition to the updated zone averages of density,
          pressure, and the velocities, an estimate of the Courant



     Page 10                                         (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          number is also returned to the calling program.  In order
          for the method to be stable, the various Domains of
          Dependence for each zone interface may not exceed one zone
          in width.  Nor may any wave travel more than one zone width
          during the timestep.  During the remap step, stability
          requires that no more than one vone (volume or mass) may be
          advected across any interface during a single timestep.


          The three-dimensional Eulerian hydrodynamic equations may be
          approximated by applying this one-dimensional operator by
          applying directional-splitting to treat gradients in the x-,
          y-, and z-directions independently.  An x-pass is performed
          first to compute the gradients in the x-direction by
          applying the one-dimensional operator to each row of the
          data domain.  This is then followed by a y-pass in which the
          one-dimensional operator is applied to each column of the
          data domain, giving the y-direction gradients. Finally a z-
          pass in which the one-dimensional operator is applied to
          each column of the data domain in the z-direction, giving
          the z-direction gradients. By applying a one-dimensional
          operator in a symmetrized order, second-order accuracy can
          be obtained.



          A gamma law equation of state is assumed:
                 p = ( gamma -1 ) * rho * ei
          where p is the pressure, rho is the density, ei is the
          internal energy, and gamma is the ratio of specific heats.
          Gamma is assumed to be constant and uniform.



     HHHHIIIINNNNTTTTSSSS
          +o    The required problem boundary conditions may be
               enforced by setting appropriate zone-averaged values of
               the fake zones.


          +o    The timestep should be adjusted so that the Courant
               number (COURMX) remains less than 1.0.  Because the
               actual Courant number encountered during a call is not
               known until after the call is completed, a common
               procedure is to adjust the new timestep so that the
               Courant number from the just completed call is <= 0.8.


          +o    To maintain the second order accuracy, the timestep
               must not be varied between the sets of symmetrized
               passes x-y-z, z-y-x, etc.  [[[[dddd____]]]]ddddoooo____ppppppppmmmmllllrrrr0000____1111ddddcccc.




     Page 11                                         (printed 2/18/99)






     DDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVVDDDDOOOO____PPPPPPPPMMMMLLLLRRRR0000____3333DDDDCCCC____GGGGAAAAMMMMMMMMAAAA((((JJJJaaaannnn  1111999999999999))))



          +o    For an example of calls to the general Equation of
               state routines, see [[[[dddd____]]]]ddddoooo____ppppppppmmmmllllrrrr0000____1111ddddcccc.


          +o    For an example of an non uniform grid see
               [[[[dddd____]]]]ddddoooo____ppppppppmmmmllllrrrr____1111ddddcccc____ggggaaaammmmmmmmaaaa.


          +o    For examples of higher dimensions see
               [[[[dddd____]]]]ddddoooo____ppppppppmmmmllllrrrr0000____1111ddddcccc____ggggaaaammmmmmmmaaaa or [[[[dddd____]]]]ddddoooo____ppppppppmmmmllllrrrr0000____2222ddddcccc____ggggaaaammmmmmmmaaaa .



     BBBBUUUUGGGGSSSS
          +o

     SSSSEEEEEEEE AAAALLLLSSSSOOOO
          DO_PPMLR0_1DC, DO_PPMLR0_1DC_GAMMA, DO_PPMLR_1DC_GAMMA,
          DO_PPMLR0_2DC_GAMMA,


     FFFFIIIILLLLEEEESSSS
               ppm98_dfns.h

               ppm98_bdys.h



     BBBBUUUUGGGG RRRREEEEPPPPOOOORRRRTTTTSSSS TTTTOOOO
          ppmlib@sapphire.lcse.umn.edu




     AAAAUUUUTTTTHHHHOOOORRRRSSSS
          PPPPaaaauuuullll RRRR.... WWWWooooooooddddwwwwaaaarrrrdddd paul@lcse.umn.edu
          BBBB.... KKKKeeeevvvviiiinnnn EEEEddddggggaaaarrrr  bke@lcse.umn.edu
          Department of Astronomy
          University of Minnesota
          Minneapolis, MN 55455 USA















     Page 12                                         (printed 2/18/99)