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 superstable. 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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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, yn, Euler

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