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at tn+1. Aside from di culties in starting them, they provide a mechanism for increasing
accuracy without much additional computation. Multistep schemes are widely used to
solve ordinary di erential equations 12] and we expect them to be of similar value when
solving partial di erential equations. Let's again begin with the model heat conduction 4.3. Multilevel Schemes 17 equation (4.1.1a) and replace the time derivative by a centered di erence approximation
to obtain
un+1 ; un;1
j + O( t2 ):
(ut)n = j
(4.3.1)
j
2t
Substituting (4.3.1) and the centered di erence approximation (2.1.9) for the second
spatial derivative into (4.1.1a) yields
n+1
n;1
n
n
n
n ; (u )n = uj ; uj ; uj ;1 ; 2uj + uj +1 + O( t2 ) + O( x2 ):
(ut )j
xx j
2t
x2
Neglecting the local discretization error and solving for Ujn+1 gives Ujn+1 = Ujn;1 + 2r(Ujn;1 ; 2Ujn + Ujn+1): (4.3.2) This centered timecentered space scheme is often called the leap frog method. Its stencil,
shown on the left of Figure 4.3.1, indicates that data is needed at the two time levels n
and n ; 1 in order to compute a solution at time level n + 1. Since initial data is only
prescribed at time level 0, the leap frog method is not self starting. An independent
method is needed to compute a solution at time level 1. Although the leap frog scheme
involves solutions at three time levels, we'll call it a twolevel method because it spans
the two time intervals (tn;1 tn) and (tn tn+1). 11
00
11
00
11
00 11
00
11
00 11
00
11
00 11
00
11
00
j1 j 11
00 n+1 n 11
00
11
00 111
000
111
000
11
00
11
00 n1
j+1 n+1 j1 j n n1
j+1 Figure 4.3.1: Computational stencils of the leap frog scheme (4.3.2) (left) and the Du
FortFrankel scheme (4.3.8) (right).
Before discussing starting techniques, let us examine the stability of (4.3.2) using the
von Neumann method. Thus, assume periodic initial data on 0 x < 2 , represent the 18 Parabolic PDEs solution of (4.3.2) as the discrete Fourier series (4.1.16), substitute (4.1.16) into (4.3.2),
and use the orthogonality condition (3.2.2) to obtain An+1 = An;1 + 2r(e;2
k
k ik=J ; 2 + e2 ik=J )An
k or An+1 = An;1 ; 4r k An
k
k
k (4.3.3a) where
;2 ik=J e
k =1; + e2
2 ik=J = 1 ; cos 2k = 2 sin2 k :
J
J (4.3.3b) The di erence equation (4.3.3a) is second order as opposed to the rstorder schemes
that we have been studying. Let us try, however, to obtain a solution having the same
form as the rstorder di erence equations. Thus, assume An = (Mk )nck
k (4.3.4) where Mk is, as usual, the ampli cation factor and ck is a constant. Substituting (4.3.4)
into (4.3.3a)
(Mk )n+1 = (Mk )n;1 ; 4r k(Mk )n:
Dividing by the common factor of (Mk )n;1 yields the quadratic equation
(Mk )2 + 4r kMk ; 1 = 0
which has the solution Mk = ;2r k p
1 + (2r k )2: (4.3.5) The two ampli cation factors correspond to the two independent solutions of the secondorder linear di erence equation (4.3.3a). In parallel with the solution of linear ordinary
di erential equations, the general solution of (4.3.3a) is the linear combination An = c+(Mk+ )n + c;(Mk; )n:
k
k
k (4.3.6) 4.3. Multilevel Schemes 19 of the two solutions. The constants ck can be determined from the initial condition prescribing A0 and the independent starting method that is used to compute A1 . However,
k
k
it is usually not necessary to explicitly determine ck .
It is a bit di cult to ascertain the properties of the two ampli cation factors Mk
from (4.3.5), so let us study their behavior when 0 < r k 1. Thus, expanding the
square root term in a Taylor's series Mk+ = 1 ; 2r k + O(r2 2 )
k Mk; = ; 1 + 2r k + O(r2 2 )]:
k Substituting this into (4.3.6) An = c+ 1 ; 2r k + O(r2 2 )]n + c;(;1)n 1 + 2r k + O(r2 2 )]n:
k
k
k
k
k (4.3.7) The rst term in (4.3.7) is an approximation of the solution of the heat conduction
problem. It is decaying with increasing n when r k is su ciently small. The second
term in (4.3.7) increases in amplitude and oscillates as n increases. The growth of the
solution can be restricted by maintaining r at a very small value however, there is no
value of r > 0 for which the leap frog scheme (4.3.2) is absolutely stable. The second
growing solution is called parasitic because, if n and/or r are large enough, it will grow to
dominate the correct solution. Parasitic solutions will always be present when a higherorder di erence scheme is used to approximate a lowerorder di erential equation. In
order to be useful, the parasitic solutions of such multilevel schemes should be bounded
well below the solution that is approximating the partial di erential equation.
The Du Fort and Frankel 7] scheme
Ujn+1 ; Ujn;1
U n ; (Ujn+1 + Ujn;1) + Ujn+1
= j;1
2t
x2
or
(1 + 2r)Ujn+1 = (1 ; 2r)Ujn;1 + 2r(Ujn;1 + Ujn+1) (4.3.8) uses centered time di erences with an averaged centered spatial di erence approximation
(Figure 4.3.1, right).
A von Neumann stability analysis reveals that the du FortFrankel scheme (4.3.8) is
unconditionally stable in L2 (cf. Problem 1 at the end of this section). This result seems 20 Parabolic PDEs to be at odds with the discussion in Section 4.1 which...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

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