Example 954 4 consider burgerss equation uxx uux 0

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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 convection-di 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 erential-Algebraic Equations. SIAM, Philadelphia, 1998. 3] K.E. Brenan, S.L Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Di erential-Algebraic 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 singularly-perturbed 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. Springer-Verlag, Berlin, second edition, 1993. 8] E. Hairer and G. Wanner. Solving Ordinary Di erential Equations II: Sti and Di erential Algebraic Problems. Springer-Verlag, 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. Springer-Verlag, Berlin, 1996. 41...
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