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Chemical Engineering Hand Written_Notes_Part_71

Chemical Engineering Hand Written_Notes_Part_71 - 144 4...

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144 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES The new value x ( n +1) is a function of only the past value of x i.e., x ( n ) . This is a non-iterative scheme. Implicit Euler method: (3.15) d x dt = x ( n + 1) x ( n ) h = F [ x ( n + 1) , t n +1 ] x ( n + 1) = x ( n ) + hF ( n + 1) , ( n = 0 , 1 , ....... , n 1) Each of the above equation has to be solved by iterative method. For example if we use successive substitution method for solving the resulting nonlinear equation(s), the algorithm can be stated as follows: Initialize: x (0) , t f , h, , N = t f /h FOR n = 1 TO n = N x (0) ( n + 1) = x ( n ) + hF [ x ( n ) , t n ] WHILE ( δ > ) x ( k +1) ( n + 1) = x ( n ) + hF [ x ( k ) ( n + 1) , t n +1 ] δ = || x ( k +1) ( n + 1) x ( k ) ( n + 1) || || x ( k ) ( n + 1) || END WHILE x ( n + 1) = x ( k ) ( n + 1) END FOR 3.2.2. Variable stepsize implementation with accuracy monitoring. One prac- tical di culty involved in the integration with fi xed stepsize is the choice of stepsize such that the approximation errors are kept small. Alternatively, a variable stepsize algorithm is implemented with error monitoring as follows.
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