Give formulas for evaluating any jacobians 22 write a

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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 Crank-Nicolson 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 one-dimensional time dependent partial di erential equations. SIAM Journal on Numerical Analysis, 23:778{795, 1986. 3] S. Adjerid and J.E. Flaherty. A moving-mesh 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. High-order 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 erential-Algebraic 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 Time-Dependent Problems. 9] J.E. Flaherty and P.K. Moore. Integrated space-time adaptive hp-re 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. Springer-Verlag, Berlin, second edition, 1993. 13] E. Hairer and G. Wanner. Solving Ordinary Di erential Equations II: Sti and Di erential Algebraic Problems. Springer-Verlag, 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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