Unformatted text preview: tervals to obtain xj;1 ; 2xj + xj+1 = ; (Wj+1 ; Wj )
_
__ j = 1 2 : : : J ; 1: The mesh trajectories and solution shown in Figure 4.3.3 used a mesh with J = 30. The
mesh is concentrated in the vicinity of the sharp front and following it with approximately
the correct velocity. Comparable accuracy with a uniform stationary mesh would require
more than 1000 elements. 4.3.3 A Posteriori Error Estimation
A priori estimates of local discretization errors involve unknown derivatives of the exact
solution. They are useful for ascertaining convergence rates, but rarely provide quantitative information. A posteriori error estimates use a computed solution to provide
quantitative information. The two common techniques to obtain a posteriori error estimates involve comparing solutions on di erent meshes and comparing solutions of different orders. We will sketch the latter technique and describe the former in greater
detail.
Suppose that we have two methods with local errors un+1 ; Ujn+1 = t c1 tp + c2 xp + O( tp+1) + O( xp+1)]
j (4.3.14a) un+1 ; Vjn+1 = t d1 tp+1 + d2 xp+1 + O( tp+2) + O( xp+2)]:
j (4.3.14b) The spatial and temporal orders have been equated for simplicity. The technique that
we are about to present works when these orders di er.
Adding and subtracting the higher ordersolution yields un+1 ; Ujn+1 = (un+1 ; Vjn+1 ) + (Vjn+1 ; Ujn+1 ):
j
j (4.3.15a) Using (4.3.14) we see that the term on the left of (4.3.15a) is t(O( tp) + O( xp)) and
the rst term on the right is t(O( tp+1) + O( xp+1)). Thus, the second term on the 28 Parabolic PDEs right must be t(O( tp) + O( xp)). Hence, we can neglect the rst term on the right
to obtain the approximation un+1 ; Ujn+1 Vjn+1 ; Ujn+1 :
j (4.3.15b) Thus, the di erence between the lower and higherorder solutions furnishes an estimate of the error of the lowerorder solution. Unfortunately, the technique provides
no error estimate of the higherorder solution, but it is common to propagate it forward
in time rather than propagating the lowerorder solution. This strategy is called local
extrapolation. It is generally acceptable in regions where solutions are smooth, but can
be dangerous near singulariteis.
Unless there are special properties of the di erence scheme that can be exploited,
the work involved in obtaining the error estimate is comparable to that of obtaining
the solution. It's possible that the higherorder method can just provide a \correction"
to the lowerorder method. In this case, the work to obtain the lowerorder solution
need not be duplicated when obtaining the higherorder solution. This is called order
embedding or pre nement. There are often special super convergence points where the
error converges at a faster rate than it does elsewhere. Knowledge of these points can
reduce the computational cost of obtaining the error estimate 4].
The technique of using solutions computed on di erent meshes to obtain an a posteriori error estimate is called Richardson's extrapolation 16] or hre nement. Suppose that
a solution has been obtained using scheme (4.3.14a). Obtain a second solution using half
the time and spatial step sizes, but starting from tn . The error of the second solution,
^
which we'll call Ujn+1 , is obtained from (4.3.14a) as
^
un+1 ; Ujn+1 = 2 2t c1( 2t )p + c2( 2x )p + O( tp+1) + O( xp+1)]
j
or
^
un+1 ; Ujn+1 = 2pt c1 tp + c2 xp + O( tp + 1) + O( xp + 1)]
j (4.3.16) Remark 1. The indexing of this second ( nemesh) solution should have been changed.
^
Thus, Ujn+1 corresponds to U2nj+2. We haven't changed the indexing to emphasize that 4.3. Multilevel Schemes 29 the two solutions are sampled at the same points. Additionally, the mesh can be changed
by other than a factor of two, but more e ort is involved.
Subtracting (4.3.16) from (4.3.14a) yields
1
^
Ujn+1 ; Ujn+1 = (1 ; 2p ) t c1 tp + c2 xp] + t(O( tp+1 ) + O( xp+1)):
Neglecting the higherorder terms yields
^ n+1 n+1
p + c xp ] Uj ; Uj :
t c1 t 2
1 ; 21p
Thus, the error of the coarsemesh solution (4.3.14a) may be estimated as
^
Ujn+1 ; Ujn+1
:
(4.3.17)
un+1 ; Ujn+1
j
1 ; 21p
Richardson's extrapolation can also be used to estimate the error of the nemesh
^
solution Ujn+1 , but only at the coarse mesh points. The work of obtaining the error
estimate is approximately four times the work of obtaining the solution. This is usually
considered excessive unless the nemesh solution is propagated forward in time. As with
order embedding, this can be risky near singularities.
Example 4.3.3. Let us solve the kinematic wave equation ut + aux = 0
by the forward timebackward space scheme (2.2.4) Ujn+1 = (1 ; )Ujn + Ujn;1:
As usual = a t= x is the Courant number. We choose a = 1,
periodic initial data
u(x 0) = sin x
Thus, the exact solution is = 1=2, and the u(x t) = sin(x ; t): Since this scheme is rst order accurate, we obtain error estimates by setting p = 1
in (4.3.17). This gives
^
un+1 ; Ujn+1 Ejn+1 = 2(Ujn+1 ; Ujn+1 ):
j 30 Parabolic PDEs Let us compare exact and estimated local errors by solving this problem for a single
time step on 0 < x < 2 using meshes with spacing x = 2 =J , J = 4 8 16. Results
for the exact errors and those estimated by Richardson's extrapolation, present...
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Full Document
 Spring '14
 JosephE.Flaherty
 The Land, Boundary value problem, Partial differential equation, Dirichlet boundary condition, Neumann boundary condition, di erences

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