Unformatted text preview: eclet number (9.5.23) used
8 U (x )
if Uj;1=2 < 0
<j
!j = : U (xj;1=2 ) if Uj;1=2 = 0 :
U (xj ; 1) if Uj;1=2 > 0
The nonlinear solution is obtained by iteration with the values of U (x) evaluated at the
beginning of an iterative step.
The results for the pointwise error
jej1 = 0max ju(xj ) ; U (xj )j
jN are shown in Table 9.5.1. The value of h= = 6 is approximately where the greatest di erence between upwind di erencing ( = sgn ) and exponential weighting ( =
coth =2 ; 2= ) exists. Di erences between the two methods decrease for larger and
smaller values of h= .
The solution of convectiondi usion problems is still an active research area and much
more work is needed. This is especially the case in two and three dimensions. Those
interested in additional material may consult Roos et al. 10]. Problems
1. Consider (9.5.5) when !(x) , q(x) > 0, x 2 0 1] 5]. 1.1. Show that the solution of (9.5.5) is asymptotically given by
p
p
(x)
u(x) f (x) ; uR(0)e;x q(0)= ; uR(1)e;(1;x) q(1)= :
q
p
Thus, the solution has O( ) boundary layers at both x = 0 and x = 1.
1.2. In a similar manner, show that the Green's function is asymptotically given
by
( ;( ;x)pq( )=
1
G( x) 2 2 q(x)q( )]1=4 e;(x; )pq( )= if x
:
e
if x >
The Green's function is exponentially small away from x = , where it has
two boundary layers. The Green's function is also unbounded as O( ;1=2 ) at
x = as ! 0. 40 Parabolic Problems Bibliography
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Phenomena Using Singular Perturbation Methods, volume 56 of Proceedings of Symposia in Applied Mathematics, pages 47{83, Providence, 1999. AMS.
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Problems in Di erentialAlgebraic Equations. North Holland, New York, 1989.
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Methods in Engineering, 18:782{791, 1982.
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Statistical Computation, 1:260{289, 1990.
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Prentice Hall, Englewood Cli s, 1971.
7] E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Di erential Equations I:
Nonsti Problems. SpringerVerlag, Berlin, second edition, 1993.
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Di erential Algebraic Problems. SpringerVerlag, Berlin, 1991.
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Di erential Equations. SpringerVerlag, Berlin, 1996.
41...
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 Spring '14
 JosephE.Flaherty
 The Land, Tn, Boundary value problem, Numerical ordinary differential equations, nite element, parabolic problems

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