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272     6 Boundary value problems



                       1

                                                     e     1  e   1
                    ϕ                          ϕ       c
                            e  c    1   e   1   2
                                           1                          1

                                                 1
                         e  c    1   e              e  c    21   e   11
                    ϕ                          ϕ
                     2
                                           1                          1

                       1                         1
                          e  c       e   2         e  c    1   e   1
                    ϕ                          ϕ
                                               −
                                           1                          1

                   Figure 6.7 CDS solution of a 1-D convection/diffusion equation for various values of the local Peclet
                   number below and above the critical value of 2 (N = 50).


                   We next define the local Peclet number
                                            v( z)         ( z)      Pe
                                     Pe loc ≡     = (Pe)        =                     (6.79)
                                                           L      N + 1

                   to write the CDS equation for grid point j as

                                   − (Pe loc + 2)ϕ j−1 + 4ϕ j + (Pe loc − 2)ϕ j+1 = 0  (6.80)

                   With α ≡ Pe loc − 2,β ≡ Pe loc + 2, the CDS linear system is

                                    4    α                             
                                                      
                                                            ϕ 1       βϕ 0
                                  −β    4   α         
                                                         ϕ 2      0  
                                                 .                     
                                                 .        .       .  
                                       −β   4     .       . .   =   . .         (6.81)
                                                      
                                             .   .                     
                                             .   .                0  
                                             .    .  α    ϕ N−1
                                                 −β   4     ϕ N      −αϕ 1
                   While for all Pe loc ≥ 0wehave β> 0, α changes sign at Pe loc = 2. We expect some
                   qualitative change in the solution when this occurs. The CDS solution is plotted in Figure 6.7
                   for various values of Pe loc for a grid of 50 points. When Pe loc >2, the numerical solution
                   exhibits oscillations that are not present in the true solution. Also, as this equation models the
                   convection and diffusion of a density field, it is physically unrealistic for ϕ to take negative
                   values, but it does so in the numerical solution when Pe loc > 2. To remove such spurious
                   oscillations, we can make the grid finer, thus decreasing Pe loc for fixed Pe. However, often
                   we do not know a priori how small to make the grid to avoid the oscillations, whose effect
                   on the convergence of numerical algorithms may be severe.
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