Chemical Engineering Hand Written_Notes_Part_31

Chemical Engineering Hand Written_Notes_Part_31 - 64 3...

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64 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES —Approach2 : (3.15) f 1 [ ( y 1 y 1 ) (2 z ) ,y 0 , 0] = 0 (3.16) f 2 [ y n +1 y n 1 ( z ) ,y 0 , 0] = 0 This approach introduces two more variables y 1 and y n +1 at hy- pothetical grid points. Thus we have n +3 variables and n +1 equations, two more algebraic equations can be generated by set- ting residual at zero at boundary points,i.e., at z 0 and z n ,i.e., R 0 =0 and R n =0 This results in ( n +3) equations in ( n +3) unknowns. Remark 1 . Forcing residual R i =0 at internal grid points implies choosing the values of y i such that the ODE-BVP is satis f ed exactly at the internal grid points. Obviously larger the value of N , the closer is the numerical solution expected to be near exact solution. Note that R i =0 at internal grid points does not imply y i equals y i ( exact solution ) at that point, i.e. y ( z i ) 6 = y ( z i ) .
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Chemical Engineering Hand Written_Notes_Part_31 - 64 3...

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