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312                                            10.  Inviscid  Compressible  Flow


         Lax-Wendroff  and  MacCormack  explicit  schemes  briefly  discussed  in  Sections
         5.2  and  5.3.  It  then  becomes  necessary  to  add  explicit  artificial  dissipation  to
         the  implicit  methods  of  Section  5.4. The  amount  of dissipation  must  be  enough
         to  damp  the  instabilities  and,  in  addition,  to  prevent  oscillations  around  shock
         waves.  The  model  problem  of  Section  10.13  will  show  how  explicit  artificial
         dissipation  is introduced  into  the  formulation  of  implicit  schemes.



         10.9  MacCormack      Method    for  Compressible    Euler   Equations

         In  Section  5.3  we  discussed  the  explicit  MacCormack  method  for  a  one-
         dimensional  problem  and  here  we  discuss  its  extension  to  compressible  flows.
         The  Euler  equations  for  one-dimensional  flow  are  considered  and  from  Eq.
         (2.1.30),



         with  Q  and  E  given  by  Eqs.  (2.2.32a)  for  one-dimensional  flow.
                                                                               / )
            The  discussion  of  Section  5.3  is  followed,  the  predictor  values  at  {xi,  £ n + 1 2
         is  defined  by  Q™  '  (=  Qi)  and  the  convective  flux  term  E  represented  with
         forward  differences  in  the  predictor  step  and  backward  differences  in  the  cor-
         rector  step.  Thus
         predictor:

                                Qi  =  Qf  -  At  [ E?+1 Ax E?  )          (10-9.2)

         corrector:
                                Q i  Q?-At(%-^A                           (10.9.3)
                                   =
         updating
                                    Qi +x  =  \(Qi  +  Qi)                 (10-9-4)

         Note  that  the  predictor  step  could  have  been  obtained  with  backward  differ-
         ences,  while  the  corrector  step  with  forward  differences  as  in  Eqs.  (5.3.3),  that
         is

         predictor:
                                                    i 1
                                                '  Ax ~ )                 (10.9.5)
         corrector:
                                                   Ei
                                Qi  =  Q?  -  At  { Ei+l Ax )              (10-9.6)

         Updating  is  as  described  by  Eq.  (10.9.4).  It  can  be  shown  that  the  two  proce-
         dures  are  identical  for  linear  equations,  while  for  non-linear  equations  they  are
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