Chemical Engineering Hand Written_Notes_Part_78

Chemical Engineering Hand Written_Notes_Part_78 - 158 4....

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158 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES Trapeziodal Rule (Simpson’s method) (4.11) " e ( n +1) x ( n # = 1+ ah + ( ah ) 2 2 e ah 1+( ah/ 2) 1 ( ah/ 2) ¸ 0 e ah " e ( n ) x ( n ) # (4.12) ¯ ¯ ¯ ¯ ah/ 2) 1 ( ah/ 2) ¯ ¯ ¯ ¯ < 1 ah < 0 2’nd Order Runge Kutta Method " e ( n x ( n # (4.13) = μ ah + ( ah ) 2 2 e ah μ ah + ( ah ) 2 2 ¶¸ 0 e ah " e ( n ) x ( n ) # (4.14) (4.15) ¯ ¯ ¯ ¯ ah + ( ah ) 2 2 ¯ ¯ ¯ ¯ < 1 ⇒− 2 <ah + ( ah ) 2 2 < 0 Thus, choice of integration interval depends on the parameters of the equa- tion to be solved and the method used for solving ODE IVP. These simple example also demonstrates that the approximation error analysis gives consid- erable insight into relative merits of di f erent methods. For example, in the case of implicit Euler or Simpson’s rule, the approximation error asymptotically re- duces to zero for any choice of h> 0 . (Of course, larger the value of h , less accurate is the numerical solution.) Same is not true for explicit Euler method.
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Chemical Engineering Hand Written_Notes_Part_78 - 158 4....

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