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Unformatted text preview: Lecture 37 Implicit Methods The Implicit Difference Equations By approximating u xx and u t at t j +1 rather than t j , and using a backwards difference for u t , the equation u t = cu xx is approximated by u i,j +1- u i,j k = c h 2 ( u i- 1 ,j +1- 2 u i,j +1 + u i +1 ,j +1 ) . (37.1) Note that all the terms have index j + 1 except one and isolating this term leads to u i,j =- ru i- 1 ,j +1 + (1 + 2 r ) u i,j +1- ru i +1 ,j +1 for 1 i m- 1 , (37.2) where r = ck/h 2 as before. Now we have u j given in terms of u j +1 . This would seem like a problem, until you consider that the relationship is linear. Using matrix notation, we have u j = B u j +1- r b j +1 , where b j +1 represents the boundary condition. Thus to find u j +1 , we need only solve the linear system B u j +1 = u j + r b j +1 , (37.3) where u j and b j +1 are given and B = 1 + 2 r- r- r 1 + 2 r- r . . . . . . . . .- r 1 + 2 r- r- r 1 + 2 r . (37.4) Using this scheme is called the implicit method since u j +1 is defined implicitly. Since we are solving (37.1), the most important quantity is the maximum absolute eigenvalue of B- 1 , which is 1 divided by the smallest ew of B . Figure 37.1 shows the maximum absolute ew s of B- 1 as a function of r for various size matrices. Notice that this absolute maximum is always lessfor various size matrices....
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- Fall '08