Substituting 9517a 9518a 9519a and 9520 into

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Unformatted text preview: + 2= + = + 2= ; 1 : (9.5.22b) 1 In order to avoid the spurious oscillations found in Example 9.5.1, we'll insist that > 0. Using (9.5.22b), we see that this requires (9.5.22c) > sgn ; 2 : Some speci c choices of follow: 9.5. Convection-Di usion Systems 37 1. Galerkin's method, = 0. In this case, ^(s) = ^(s) = 1 ; jsj : 2 Using (9.5.22), this method is oscillation free when 2 > 1: jj From (9.5.14c), this requires h < 2j =!j. For small values of j =!j, this would be too restrictive. 2. Il'in's scheme. In this case, ^(s) is given by (9.5.14b) and = coth 2 ; 2 : This scheme gives the exact solution at element vertices for all values of . Either this result or the use of (9.5.22c) indicates that the solution will be oscillation free for all values of . This choice of is shown with the function 1 ; 2= in Figure 9.5.3. 3. Upwind di erencing, = sgn . When > 0, the shape function ^(s) is the piecewise constant function ^(s) = 1 if ; 1 < s 0 : 0 otherwise This function is discontinuous however, nite element solutions still converge. With = 1, (9.5.22b) becomes + = 2(1 2= 1= ) : In the limit as ! 1, we have thus, using (9.5.22a) ci 1 ; ;(N ;i) i = 0 1 ::: N 1: This result is a good asymptotic approximation of the true solution. Examining (9.5.21) as a nite di erence equation, we see that positive values of can be regarded as adding dissipation to the system. This approach can also be used for variable-coe cient problems and for nonuniform mesh spacing. The cell Peclet number would depend on the local value of ! and the mesh spacing in this case and could be selected as j = hj !j (9.5.23) 38 Parabolic Problems 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Figure 9.5.3: The upwinding parameter = coth =2 ; 2= for Il'in's scheme (upper curve) and the function 1 ; 2= (lower curve) vs. . where hj = xj ; xj;1 and !j is a characteristic value of !(x) when x 2 xj;1 xj ), e.g., !j = !j+1=2. Upwind di erencing is too di usive for many applications. Il'in's scheme o ers advantages, but it is di cult to extend to problems other than (9.5.5). The Petrov-Galerkin technique has also been applied to transient problems of ther form (9.5.1) however, the results of applying Il'in's scheme to transient problems have more di usion than when it is applied to steady problems. Example 9.5.4 4]. Consider Burgers's equation uxx ; uux = 0 0<x<1 with the Dirichlet boundary conditions selected so that the exact solution is u(x) = tanh 1 ; x : Burgers's equation is often used as a test problem because it is a nonlinear problem with a known exact solution that has a behavior found in more complex problems. Flaherty 4] solved problems with h= = 6 500 and N = 20 using upwind di erencing and Il'in's scheme (the Petrov-Galerkin method with the exponential weighting given by (9.5.14b)). 9.5. Convection-Di usion Systems 39 h= Maximum Error Upwind Exponential 6 0.124 0.0766 500 0.00200 0.00100 Table 9.5.1: Maximum pointwise errors for the solution of Example 9.5.4 using upwind di erencing ( = sgn ) and exponential weighting ( = coth =2 ; 2= ) 4]. The cell P...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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