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. ConvectionDi 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 variablecoe 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 PetrovGalerkin 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 PetrovGalerkin method with the exponential weighting given by (9.5.14b)). 9.5. ConvectionDi 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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 Spring '14
 JosephE.Flaherty
 The Land, Tn, Boundary value problem, Numerical ordinary differential equations, nite element, parabolic problems

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