Page 171 - Basic Structured Grid Generation
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160  Basic Structured Grid Generation

                        particular where the first derivative u x (= du/dx) has a high value so that grid-points
                        locally are close together. The most commonly used weight functions are
                                                     
                                                     u x (x)
                                                     
                                                     

                                                             2
                                             W (x) =    1 + α (u x ) 2                     (6.24)
                                                     
                                                     
                                                      (1 + βκ) 1 + α (u x ) ,
                                                                   2    2
                        similar to the expressions given in eqns (2.58), (2.59), and (2.60), where here κ is the
                        curvature of the solution, given by
                                                            |u xx |
                                                    κ =                                    (6.25)
                                                                2 3/2
                                                        [1 + (u x ) ]
                        as in eqn (2.17), and α, β are parameters to be chosen by the user.
                          Note that with W(x) = u x , eqn (6.21) becomes
                                                     du dx   du
                                                          =     = K,                       (6.26)
                                                     dx dξ   dξ
                        which has the consequence that the increments in the values of u between the equally-
                        spaced points on the ξ interval and between the corresponding unequally-spaced grid
                        points in the x interval remain the same. Figure 6.1 shows an example where a decreas-
                        ing gradient of u with x results in a higher density of grid points at low values of x
                        where u x is higher.
                          This choice of weight function may have the disadvantage that the grid-point spacing
                        is too large when u x is small. An alternative is

                                                    W(x) =   1 + (u x ) 2                  (6.27)
                        with the particular choice of α = 1 in (6.24), which approximates to u x when u x is
                        large. This choice can be neatly interpreted in the ux plane in terms of the distance s
                        between points on the solution curve, since

                                                 2      2           2
                                       ds =  (du) + (dx) =   1 + (u x ) dx = W(x) dx,
                        so that eqn (6.21) becomes
                                                     ds dx   ds
                                                          =     = K.                       (6.28)
                                                     dx dξ   dξ


                                               u
                                                                     i +2
                                                               i +1
                                                            i





                                              O
                                                   a                     x
                        Fig. 6.1 Adaptation in one dimension.
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