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Table 4.3.3, indicate no di erences between the two. Indeed, the estimated errors are
so accurate for this problem that when added to the computed solution with J = 4
they produce a maximum error of 0:2220 10;15. Results with less smooth solutions
would not normally be this good, but, nevertheless, there are decided advantages to
using Richardson's extrapolation.
J ku1 ; U1k1 kE1k1
Table 4.3.2: Exact and estimated local errors for Example 4.3.3. Problems
1. ( 15], p. 91.) Let us consider the Du Fort-Frankel scheme (4.3.8) applied to the
heat conduction equation (4.1.1a).
1.1. Show that (4.3.8) satis es the von Neumann necessary condition for stability
for all r > 0.
1.2. Calculate the leading terms of the local discretization error of (4.3.8) and
examine the scheme's consistency. If the scheme is conditionally consistent,
clearly state the conditions when it is and is not consistent.
2. ( 15], p. 45.) Derive the following \summation by parts" formulas assuming that
a0 = aJ = b0 = bJ = 0.
2.2. J ;1
j =1 aj (bj ; bj;1) = ; aj (bj+1 ; bj ) = ; J ;1
j =1 J ;1
j =1 bj (aj+1 ; aj ): bj+1(aj+1 ; aj ) ; a1 b1 : 4.4. Variable-Coe cient and Nonlinear Problems
2.3. J ;1
j =1 aj (bj+1 ; 2bj + bj;1 ) = ; J ;1
j =1 31
(bj+1 ; bj )(aj+1 ; aj ) ; a1 b1 : 4.4 Variable-Coe cient and Nonlinear Problems
The methods presented in the preceding sections and chapters can be applied to variablecoe cient and nonlinear problems. Variable-coe cient problems pose virtually no computational di culties. Likewise, explicit nonlinear problems lead to simple algebraic
problems however, implicit schemes will generally require an iterative solution technique.
Analyses of consistency, convergence, and stability are more di cult for both variablecoe cient and nonlinear problems than they are for the constant-coe cient cases that
we have been studying. If coe cients and/or solutions are smooth, one can often invoke
a \local analysis," which is a constant-coe cient analysis based on local values of the
coe cients. 4.4.1 Linear Variable-Coe cient Problems
Let us begin with some examples.
Example 4.4.1. Consider the heat conduction problem ut = uxx + aux + bu + f (4.4.1) where , a, b, and f are functions of x and t. The explicit forward time-centered space
scheme for this problem would be
Ujn+1 ; Ujn n 2Ujn n Ujn n n n
t = j x2 + aj x + bj Uj + fj
where jn = (j x n t), etc. on a uniform mesh of spacing ( x t). Solving, as usual,
for Ujn+1 using the de nitions (Table 2.1.1) of the averaging and central di erence
operators, we nd Ujn+1 = c;1Ujn;1 + c0 Ujn + c1Ujn+1 + tfjn
where (4.4.2a) xan
c;1 = ( jn ; 2 j ) xt2 (4.4.2b) 32 Parabolic PDEs c0 = 1 ; (2 jn ; x2 bn) xt2
j (4.4.2c) xan t
c1 = (
2 ) x2 :
As noted, this scheme is used in exactly the same manner as described for the constantcoe cient problems studied earlier. Questions of consistency, convergence, and stability
will be addressed shortly.
Example 4.4.2. Crank-Nicolson schemes for (4.4.1) o er some possibilities for variation. One possibility is to center the coe cients at (j n + 1=2) and average the solution
Ujn+1 ; Ujn
n+1=2 ]( Uj + Uj ) + f n+1=2 :
x2 + aj
x + bj
The symmetry of the discrete operator about (n + 1=2) t will give an O( t2) local error
in time. Centering about j x, as in the past, gives an O( x2) local spatial error.
A second Crank-Nicolson scheme can be obtained by averaging the spatial operator
on the right side of (4.4.1) to obtain
Ujn+1 ; Ujn 1 n+1 2
= 2 ( j x2 + an+1 x + bn+1 )Ujn+1 + fjn+1]
+2 ( n
j 2 n
x2 + aj x + bj )Uj + fj ]
The form (4.4.3a) is analogous to midpoint quadrature while (4.4.3b) is analogous to
the trapezoidal rule.
Example 4.4.3. The di usion term in the heat conduction equation frequently appears
in the self-adjoint form ( ux)x. Problems in cylindrical coordinates, for example, would
have the form ((1=x)ux)x. Such expressions could clearly be expanded to uxx + x ux
and approximated as in Example 4.4.1 however, one would have to know x. A better
alternative is to approximate the self-adjoint di usive term directly. For example, using
central di erences yields
@ ( @u )jn ( jn Ujn ) = jn+1=2 Ujn+1 ; ( jn+1=2 + jn;1=2 )Ujn + jn;1=2Ujn;1 : (4.4.4)
@x @x j
x2 4.4. Variable-Coe cient and Nonlinear Problems 33 The consistency, convergence, and stability of these variable-coe cient di erence
schemes must be established. In order to simplicity this discussion, let us suppose that
(4.4.1) is solved by a general explicit nite di erence scheme of the form
csUj+s + tfjn :
js j S The coe cients cs, jsj S , can depend on j , n, x, and t. A similar development is
possible for implicit schemes.
Recall that (cf. De nitions 3.1.2, 3) consistency implies that the local error
un+1 ; Ujn+1 = ; t jn = un+1 ;
csuj+s ; tfjn
jsj S tend to zero faster than O( t). Expanding (4.4.6) in a Taylor's series about...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14
- The Land