Biomedical Engineering Reference
In-Depth Information
additional constant, C i , so that the approximation
γ i 1 u i 1 + γ i u i + γ i + 1 u i + 1 + κ i u i = f i + C i
(4.62)
is second-order accurate, with jump conditions Eqs. 4.60 and 4.61.
To determine the γ i 's, Taylor expansions are taken about the point x = α to
get
1
2 ( x i 1 α ) 2 u xx + O ( x 3 ) ,
u ( x i 1 ) = u + ( x i 1 α ) u x +
(4.63)
1
2 ( x i α ) 2 u xx + O ( x 3 ) ,
u ( x i ) = u + ( x i α ) u x +
(4.64)
1
2 ( x i + 1 α ) 2 u xx + O ( x 3 ) .
u ( x i + 1 ) = u + + ( x i + 1 α ) u x +
(4.65)
These expansions are inserted into Eq. 4.62, and the u + terms are eliminated
from the equation by using the jump conditions Eqs. 4.60 and 4.61, combined
with the equation
( β u x ) x + κ u + = ( β u x ) x + κ u ,
(4.66)
which comes from the continuity of f in Eq. 4.59. The function f on the right side
of Eq. 4.62 is replaced with the approximation from the left side, f = ( β u x ) x +
κ u . This results in the following equation:
γ i 1 u + ( x i 1 α ) u x +
2 ( x i 1 α ) 2 u xx
1
+ γ i u + ( x i α ) u x +
2 ( x i α ) 2 u xx
+ γ i + 1 u + a + ( x i + 1 α )( u x + b ) +
1
2 ( x i + 1 α ) 2 u xx
1
b β x κ a
β
2 ( x i α ) 2 u xx
= β x u x + β u xx + κ u + C i + O ( x 3 )
u + ( x i α ) u x +
1
+ κ
(4.67)
The coefficients γ i 1 , γ i , γ i + 1 , and C i are now chosen so that Eq. 4.67 holds up
to second order. This leads to the following equations:
γ i 1 + γ i + γ i + 1 = 0 ,
(4.68)
γ i 1 ( x i 1 α ) + γ i ( x i α ) + γ i + 1 ( x i + 1 α ) + κ ( x i α ) = β x ,
(4.69)
γ i 1 ( x i 1 α ) 2
+ γ i ( x i α ) 2
+ γ i + 1 ( x i + 1 α ) 2
+ κ ( x i α ) 2
= 2 β,
(4.70)
γ i + 1 a + b ( x i + 1 α )
( b β x κ a )( x i + 1 α ) 2
β
1
2
= C i .
(4.71)
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