1 02 03 04 05 t 06 07 08 09 figure 222 euler

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Unformatted text preview: 0) = (2.2.1) y0 : As with the explicit Euler method, we'll expand the exact solution in a Taylor's series however, this time, let us expand about the time n to get t ( ) = ( n) ; 2 ( n ) + 2 ( n) (2.2.2) n1 n n The backward Euler method is obtained by using (2.2.1) to eliminate and neglecting the remainder term in the Taylor series. Thus, y tn;1 yt hy 0 h t y 00 t ; < <t : y yn = yn;1 + ( 0 ) (2.2.3) hf tn yn : The method is called implicit because (2.2.3) only determines the solution when we can solve a (generally) nonlinear algebraic equation for n. Before discussing solutions, however, let us examine the region of absolute stability in order to determine whether or not the backward Euler method has better stability properties than the forward Euler method. Hence, let us apply (2.2.3) to the test equation (2.1.15) to obtain y yn = yn;1 + yn : h Although implicit, (2.2.3) is simply solved for the (2.1.15) to yield yn;1 = 1; As noted in Section 2.1, since problem (2.1.15) is linear we can regard n as a perturbation and impose the absolute stability condition to n itself. In this case, n will not grow when 1 j1 ; j 1 yn : h y y y h or j1 ; j 1 h : (2.2.4) Thus, the region of absolute stability of the backward Euler method is the region outside of a unit circle centered at (1 0) in the complex plane (Figure 2.2.3). h 22 Im(h λ) 1 -1 1 Re(h λ) 2 -1 Figure 2.2.3: Region of absolute stability (shaded) of the backward Euler method. The test equation (2.1.15) is asymptotically stable in the entire left half of the complex plane and stable on the imaginary axis. When approximated by the backward Euler method, it is not only absolutely stable there, but is also absolutely stable in most of the right half of the plane. Methods, such as the backward Euler method, that are stable when the di erential equation is not are called super-stable. We'll have to see what this means. Local errors of the backward Euler method are only slightly more di cult to estimate than for the explicit Euler method. Using (2.2.2) and (2.2.3), for example, we obtain the local discretization error as ( n) ; ( n 1) ; ( ( )) = ; ( ) (2.2.5) n= n n n1 n n 2n The local error satis es h yt yt ; ft h dn = ( n) ; yt h yt yn = h n + 00 t = ( n) ; ( n 1 ) ; yt yt or dn y ; ( )) ; ( n ( h f tn y tn ft ; < ( hf tn yn yn ) )] : Using the mean value theorem ( ( )) ; ( n f tn y tn ft yn ) = y( n f 23 t n )( ( n) ; n) yt y <t : := := 0 (0) yn yn;1 repeat := n 1 + := + 1 until converged ( +1) n := n ( +1) yn y y ; ( () hf tn yn ) y Figure 2.2.4: Functional iteration to obtain method. where n is between yn yn from yn;1 using the backward Euler and ( n). Thus, yt dn = h n + ( hfy tn n ) dn or, using (2.2.5), ( 2 2) ( n) n=; 1 ; y ( n n) h= d y hf 00 t (2.2.6) : The relationship between the local error n and the local discretization error n is not as simple as for explicit di erence equations. However, an examination of (2.1.4) reveals that the local errors of the explicit and implicit Euler methods are both ( 2). Since n is de ne...
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