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Unformatted text preview: (j n) gives
22
X
; t jn = un + t(ut )n + O( t2 ) ; cs un + s x(ux)n + s 2 x (uxx)n + O( x3 )] ; tfjn :
j
j
j
j
j
jsj S
Let us use the di erential equation (4.4.1) to eliminate ut in favor of spatial derivatives
to obtain ; t jn = un + t jn (uxx)n + an(ux)n + bnun + fjn] + O( t2)
j
j
j
j
jj
; X jsj Regrouping terms 22
cs un + s x(ux)n + s 2 x (uxx)n + O( x3)] ; tfjn:
j
j
j
S ; t jn = 1 ;
; x2
Enforcing that n
j X
jsj S cs + tbn ]un + ; x
jj X
jsj S scs + tan ](ux)n+
j
j X s2 cs
+ t jn ](uxx)n + O( t2) + O( x3):
j
jsj S 2 ! 0 as x t ! 0 yields the consistency conditions
X
lim!0 1t
cs ; 1] = bn
j
xt
jsj S (4.4.7a) X
n
lim!0 x
xt
t jsj S scs = aj (4.4.7b) 34 Parabolic PDEs
2 X s2 cs
n
lim!0 xt
xt
2 = j:
jsj S (4.4.7c) Additional terms in the Taylor's series expansion may give rise to other consistency
conditions if, e.g., mesh ratios such as x3 = t did not approach zero. However, as we
have seen, it's usually necessary to require x2 = t to be bounded for an explicit scheme.
Thus, one would hardly compute with the smaller t = O( x3).
Stability is usually governed by the highest order terms in the di erence equation.
Let us demonstrate this by performing a von Neumann analysis of the explicit scheme
(4.4.5) assuming that coe cients are constant (cs, jsj S , are independent of j and n),
fjn = 0, and computation is performed with constant t= x2 . Under these restrictions,
the von Neumann analysis produces the ampli cation factor
Xk
Mk =
cs!s
!s = e2 is=J :
(4.4.8)
jsj S Suppose the coe cients cs, jsj
powers of t1=2 of the form S , are smooth enough to have series expansions in cs = c0 + t1=2 c1=2 + tc1 + : : : :
s
s
s (4.4.9) Substituting (4.4.9) into the consistency conditions (4.4.7) yields an alternate set of
consistency conditions having the form
X0
X 1=2
X1
cs = 1
cs = 0
cs = b
(4.4.10a)
jsj S X0
scs = 0 jsj S jsj S jsj S r x2 X sc1=2 = a
t jsj S s x2 X s2c0 = :
s
t jsj S 2 (4.4.10b)
(4.4.10c) The subscript j and superscript n on the coe cients a, b, and have been omitted for
this constant coe cient analysis.
Remark 1. We encountered the rst consistency condition in (4.4.10) in conjunction
with the Maximum Principle (Theorem 3.1.1). 4.4. VariableCoe cient and Nonlinear Problems 35 Substituting the expansions (4.4.9) into (4.4.8) yields an expansion of the ampli cation factor as Mk = Mk0 + t1=2 Mk1=2 + tMk1 + : : : (4.4.11a) where Mkl = X
jsj S k
cls!s l = 0 1 ::: : (4.4.11b) Using (4.4.10), we see that the leading term Mk0 is independent of a and b. Thus, Mk0 is
the ampli cation factor of the di erential equation ut = uxx:
Suppose that the principal part of the operator is stable, i.e., there exists a constant m0
k
such that
jMk0j 1 + m0 t:
k
k
For simplicity, let us assume that S is small relative to J and expand !s in a series.
Using (4.4.10) and (4.4.11b), this gives t
Mk0 = 1 ; x2 ( 2J k )2 + O(( 2 JkS )3)
r t ( 2 k ) + O(( 2 kS )3):
x2 J
J
An expansion of Mk1 is not be needed. These equations imply that Mk0 is real and Mk1=2
is imaginary. Assuming this to be the case and using (4.4.11a) gives
M 1=2
k = ia jMk j = (jMk0 j2 + tjMk1=2 j2 + : : : )1=2 :
Expanding the square root jMk j 1 + O( t): Thus, the di erence scheme (4.4.5) satis es the von Neumann condition and is stable
according to Theorem 3.2.1. The proof of stability when Mk0 and Mk1 are complex follows
similar lines 17]. 36 Parabolic PDEs The previous discussion quantify the e ects of lowerorder terms on stability when the
coe cients are constant. Richtmyer and Morton 17], Chapter 5, show that a constant
coe cient stability analysis with local values of the coe cients is su cient for the stability
of a variable coe cient problem when (i) (4.4.5) is consistent and (ii) the coe cients cs,
jsj S , are continuous and uniformly bounded on the domain of the di erential equation
as t x ! 0. The local constantcoe cient analysis requires that stability be checked
for all values of the coe cients in the domain. Consider, for example, solving a heat
conduction problem with a variable di usivity (x) for x 2 (0 1). Using the explicit
forward timecentered space scheme, we would have to show that
)
1
r = (xx2 t 2
8x 2 (0 1): 4.4.2 ConvectionDi usion Problems The prior arguments imply that the lowerorder terms in a parabolic di erential equations
do not appreciably a ect stability however, caution is needed when absolute stability is
required. Consider the convectiondi usion equation ut = uxx + aux (4.4.12) which arises throughout uid mechanics. The equation clearly has properties of both the
kinematic wave equation (convection) and the heat equation (di usion) and we would
expect it to behave more like one or the other depending on the relative sizes of a and .
Let's discretize this equation by the forward timecentered space di erence approximation
to obtain Ujn+1 = Ujn + r(Ujn;1 ; 2Ujn + Ujn+1) + 2 (Ujn+1 ; Ujn;1) (4.4.13) where is the Courant number.
It is not hard to show that solutions of initial value problems for (4.4.12) with periodic
or compact data do not grow in time. Thus, it is reasonable to require kUnk kU0k:
Stability analyses involving only the hig...
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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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