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Unformatted text preview: implied that stability restrictions
were necessary for explicit nite-di erence approximations of the heat equation. The
answer to the dilemma in this case is linked to a loss of consistency unless t ! 0 at
a faster rate than x (again, cf. Problem 1 at the end of this section). Should the
computational mesh be re ned with t= x = (a constant), then the du Fort-Frankel
scheme (4.3.8) is consistent with the hyperbolic equation ut ; uxx + 2u tt = 0: Thus, the time step of the du Fort-Frankel scheme must still be restricted to be O( x2)
however, now for reasons of consistency rather than for stability.
Multistep schemes can be started with a one-step method. We must be careful to
preserve accuracy of the multistep scheme when the one-step method has a lower order.
Suppose, for example, that the time step of a second-order multistep method is tM ,
then the time step of the rst-order starting method should be O( t2 ) in order to keep
the local errors in balance. This, however, is not the case when the forward time-centered
space scheme (4.1.2) is used as a starting method for the du Fort-Frankel (4.3.8) scheme.
Although the order of (4.1.2) is O( t) + O( x2 ) and that of (4.3.8) is O( t2) + O( x2 ),
both schemes restrict t to O( x2 ). Thus, both schemes have O( x2) accuracy. 4.3.1 Matrix Stability Analysis
When using matrix methods to analyze the stability of a multilevel scheme it is convenient
to write the scheme as an equivalent one-level scheme. This is easily done and we'll
illustrate it for the du Fort-Frankel scheme (4.3.8). For simplicity, consider an initialboundary value problem for (4.3.8) with homogeneous Dirichlet boundary conditions,
then the vector form of (4.3.8) is Un+1 = AUn;1 + BUn
A = 1 ; 2r I
61 0 1
B = 1 +r2r 6
Un = 6
UJ ;1 3
5 (4.3.9b) 4.3. Multilevel Schemes
Rewrite (4.3.9a) as 21 Un+1 = B A
I0 Un :
Un;1 In this form, the multilevel du Fort-Frankel scheme looks like the one-level scheme Wn+1 = L Wn (4.3.9c) with Un
Wn = Un;1 B
0 (4.3.9d) Thus, matrix stability analysis and techniques described in Section 3.3 are directly applicable to multilevel di erence methods using the matrix L of (4.3.9d). 4.3.2 The Method of Lines
Excellent software is available for solving ordinary di erential equations. Modern multistep methods use codes containing either Adams or backward di erence formulas 5,
12, 13] to integrate the system in time. Adams methods are preferred for nonsti problems while backward di erence methods are useful for sti problems. In order to use the
ordinary di erential equations software to solve partial di erential equations of the form ut = Lu (4.3.10a) we introduce a spatial grid (as shown in Figure 4.3.2) and replace all of the spatial
derivatives appearing in the operator L by nite di erence approximations to obtain
V = L xV (4.3.10b) where L x is the discrete approximation of L and (_) denotes time di erentiation. The
elements Vj (t) of V(t) are approximations of u(xj t). Let us illustrate the idea for a heat
Example 4.3.1. The spatial operator for the heat conduction equation (4.1.1a) is
Lu = uxx. Discretizing the second spatial derivative using centered di erences gives
Vj (t) = Vj;1(t) ; 2Vjx2 ) + Vj+1(t) : 22 Parabolic PDEs
t x, j
0 1 2 j J-1 J Figure 4.3.2: Spatial discretization to be used with the method of lines.
Consider a problem with homogeneous Dirichlet boundary conditions and let V(t) =
V1(t) V2 (t) : : : Vj;1(t)]T , then the matrix form of this problem is
6 1 ;2 1
L x = x2 6
The above arguments and example emphasize that the partial di erential equation has
been \reduced," by spatial discretization, to a system of ordinary di erential equations.
The semi-discrete system (4.3.10b) can be integrated in time by most ordinary di erential
equations software. This software automatically selects time steps to satisfy a prescribed
(local) temporal error tolerance and to maintain stability. Most multistep codes also
adjust the order of accuracy of the formulas in order to improve performance. A user of
the software would only have to provide a temporal error tolerance, initial values V(0),
and a procedure for de ning the spatially-discrete operator L xV.
For heat conduction problems, the small divisor ( x2 ) present in L x suggests that
the ordinary di erential equations (4.3.10b) will be sti . These notes are not the appropriate place to initiate an extended discussion of \sti ness." Let us simply state that sti
systems have more stringent stability restrictions than nonsti systems. The accepted 4.3. Multilevel Schemes 23 S1 2
0 1 2/3 6/11
0 1 4/3 18/11
-10/147 Table 4.3.1: Coe cients of backward di erence formuas (4.3.11) 11].
remedy is to use implicit ordinary di erential equations software and the backward difference codes seem to be adequate for most problems 5, 11, 13].
The simplest backward di erence method is the implicit or backward Euler method
Vn+1 = V...
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