Unformatted text preview: on of x for each p and t= x.
Discuss the results.
2.3. Show that the discrete L2 norm J ;1
X n2
(Uj )
2= kUnk2
satis es j =0 kUn+1k2 kUnk2 when p = 2=3 and the boundary data is trivial. Also show that kUn+1k2 = kUnk2
is conserved in this case when = 0. (Hints: Multiply the CrankNicolson
equation by (Ujn+1 +Ujn ), sum over j , and use the summation by parts formulas
of Problem 4.3.2. You may also consult Mitchell and Gri ths 15], Section
2.7 and Richtmyer and Morton 17], Section 6.3.) 48 Parabolic PDEs Bibliography
1] S. Adjerid, A. Ai a, J.E. Flaherty, J.B. Hudson, and M.S. Shephard. Modeling
and adaptive numerical techniques for oxidation of ceramic composites. Ceramic
Engineering and Science Proceedings, 18:315{322, 1997.
2] S. Adjerid and J.E. Flaherty. A moving nite element method with error estimation
and re nement for onedimensional time dependent partial di erential equations.
SIAM Journal on Numerical Analysis, 23:778{795, 1986.
3] S. Adjerid and J.E. Flaherty. A movingmesh nite element method with local
re nement for parabolic partial di erential equations. Computer Methods in Applied
Mechanics and Engineering, 55:3{26, 1986.
4] S. Adjerid, J.E. Flaherty, P.K. Moore, and Y.J. Wang. Highorder adaptive methods
for parabolic systems. Physica D, 60:94{111, 1992.
5] U.M. Ascher and L.R. Petzold. Computer Methods for Ordinary Di erential Equations and Di erentialAlgebraic Equations. SIAM, Philadelphia, 1998.
6] M. Berzins and R. Furzeland. A user's manual for sprint  a versatile software
package for solving systems of algebraic, ordinary and partial di erential equations:
Part 1  algebraic and ordinary di erential equations. Technical report, Thorton
Research Centre, Shell Research Ltd., Amsterdam, 1985.
7] E.C. du Fort and S.P. Frankel. Stability conditions in the numerical treatment of
parabolic di erential equations. Mathematical Tables and Other Aids to Computation, 7:135{152, 1953.
49 50 Parabolic PDEs 8] G. Fairweather and I. Gladwell, editors. Applied Numerical Mathematics, volume 20,
1996. Special Issue on the Method of Lines for TimeDependent Problems.
9] J.E. Flaherty and P.K. Moore. Integrated spacetime adaptive hpre nement methods for parabolic systems. Applied Numerical Mathematics, 16:317{341, 1995.
10] P.R. Garabedian. Partial Di erential Equations. John Wiley and Sons, New York,
1964.
11] C.W. Gear. Numerical Initial Value Problems in Ordinary Di erential Equations.
Prentice Hall, Englewood Cli s, 1971.
12] E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Di erential Equations I:
Nonsti Problems. SpringerVerlag, Berlin, second edition, 1993.
13] E. Hairer and G. Wanner. Solving Ordinary Di erential Equations II: Sti and
Di erential Algebraic Problems. SpringerVerlag, Berlin, 1991.
14] E. Isaacson and H.B. Keller. Analysis of Numerical Methods. John Wiley and Sons,
New York, 1966.
15] A.R. Mitchell and D.F. Gri ths. The Finite Di erence Method in Partial Di erential Equations. John Wiley and Sons, Chichester, 1980.
16] L.F. Richardson and J.A. Gaunt. The deferred approach to the limit. Trans of the
Royal Society of London, 226A:299{361, 1927.
17] R.D. Richtmyer and K.W. Morton. Di erence Methods for Initial Value Problems.
John Wiley and Sons, New York, second edition, 1967....
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