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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
1] S. Adjerid, M. Ai a, and J.E. Flaherty. Computational methods for singularly perturbed systems. In J.D. Cronin and Jr. R.E. O'Malley, editors, Analyzing Multiscale
Phenomena Using Singular Perturbation Methods, volume 56 of Proceedings of Symposia in Applied Mathematics, pages 47{83, Providence, 1999. AMS.
2] U.M. Ascher and L.R. Petzold. Computer Methods for Ordinary Di erential Equations and Di erentialAlgebraic Equations. SIAM, Philadelphia, 1998.
3] K.E. Brenan, S.L Campbell, and L.R. Petzold. Numerical Solution of InitialValue
Problems in Di erentialAlgebraic Equations. North Holland, New York, 1989.
4] J.E. Flaherty. A rational function approximation for the integration point in exponentially weighted nite element methods. International Journal of Numerical
Methods in Engineering, 18:782{791, 1982.
5] J.E. Flaherty and W. Mathon. Collocation with polynomial and tension splines for
singularlyperturbed boundary value problems. SIAM Journal on Scie3nti c and
Statistical Computation, 1:260{289, 1990.
6] C.W. Gear. Numerical Initial Value Problems in Ordinary Di erential Equations.
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.
8] E. Hairer and G. Wanner. Solving Ordinary Di erential Equations II: Sti and
Di erential Algebraic Problems. SpringerVerlag, Berlin, 1991.
9] C. Johnson. Numerical Solution of Partial Di erential Equations by the Finite Element method. Cambridge, Cambridge, 1987.
10] H.G. Roos, M. Stynes, and L. Tobiska. Numerical Methods for Singularly Perturbed
Di erential Equations. SpringerVerlag, Berlin, 1996.
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

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