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10.8  Stability  Analysis  of the  Euler  Equations                   311




                                    Rloo  =  Uoc   ^                       (10.7.2)
                                                 7 - 1
         For  a  subsonic  inflow,  we  set  R\  and  s  to  their  freestream  values.  i?2  repre-
         sents  the  numerical  boundary  condition  extrapolated  from  the  interior  of  the
         computational  domain.  The  velocity  u  is  found  by  matching  R\  and  R2  on  the
         boundary.  For  a  subsonic  outflow,  R2  is  assigned  its  freestream  value,  and  s
         and  R\  are  extrapolated  from  the  interior  of  the  computational  domain.  The
         velocity  is  obtained  as  discussed  above.  Application  of  this  procedure  is  given
         in  Section  10.11.
            Along  a  solid  wall,  the  impermeable  wall  boundary  condition  is  imposed
                                         V  -n  =  0                       (10.7.3)

         and  the  density  and  pressure  are  extrapolated  from  the  interior.  Other  choices
         may  be  made  for  the  specification  of  boundary  conditions  and  the  reader  is
         referred  to  [1]  for  more  information.



         10.8  Stability  Analysis  of the  Euler   Equations

         The  stability  analysis  of  Section  5.7  indicates  that  for  the  scalar  wave  equa-
         tion,  the  maximum  time  step  allowable  in  the  explicit  Lax  method  requires  the
         CFL  condition,  CFL  <  1.  The  time  step  restriction  disappears  in  the  implicit
         schemes.  In  this  section,  we  address  the  stability  of the  Euler  equations.
            Since  the  Euler  equations  can  be  transformed  into  a  system  of  three  scalar
         equations  of the  form  given  by  Eq.  (5.1.19), the  stability  bounds  for  each  of  the
         three  equations  are  thus,  for  the  explicit  Lax  scheme

                                  u At  I Ax  <  1 for  A =  u           (10.8.1a)

                              (u  +  c)At/Ax  <  1  for  A =  u  +  c    (10.8.1b)
                              {u -  c)At/Ax  <  1 for  A =  u  -  c      (10.5.1c)
         It  is  reasonable,  although  not  mathematically  strict,  to  choose  the  minimum
         time  step  of the  three  equations,  that  is

                                CFL  -  (\u\  +  c)At/Ax  <  1            (10.5.3)
         Note  that  this  condition  must  be  satisfied  at  each  computational  point,  and
        the  time  step  must  be  taken  as the  smallest  one  in  the  computational  domain.
         Such  a  system  is referred  to  as being  numerically  stiff,  as there  is  a  wide  spread
         between the  smallest  and  largest  eigenvalues  as the  flow velocity  u departs  from
         c.
            The  analysis  is more  complicated  for  implicit  methods.  It  can  be  shown  that
         implicit  schemes  have  a  weak  instability  at  sonic  lines,  which  is damped  in  the
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