Chemical Engineering Hand Written_Notes_Part_38

Chemical Engineering Hand Written_Notes_Part_38 - 78 3...

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78 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES factor ω > 1 and get (4.29) e y y = ω ( b y y ) (4.30) or e y = ω b y + (1 ω ) y This ampli fi cation process is an extrapolation and is an example of over- relaxation . If the intermediate value b y tends to overshoot target y , then we may have to use ω < 1 ; this is called under-relaxation . Application of over-relaxation to Gauss-Seidel method leads to the fol- lowing set of equations ˜ x ( k +1) i = x ( k ) i + ω [ x ( k +1) i x ( k ) i ] (4.31) i = 1 , 2 , ..... n where x ( k +1) i are generated by Gauss-Seidel method, i.e., x ( k +1) i = μ 1 a ii " b i i 1 X j =1 a ij x ( k +1) j n X j = i +1 a ij x ( k ) j # (4.32) i = 1 , 2 , ..... n With some algebraic manipulations, the above equations can be rearranged in vector matrix form as follows (4.33) ( D + ωL ) x ( k +1) = [(1 ω ) D ωU ] x ( k ) + ω b Relaxation Algorithm INITIALIZE : b , A, x , k max , ε, ω k = 0 δ = 100 ε WHILE [( δ > ε ) AND ( k < k max )] FOR i = 1 : n q i = b i n P j =1 a ij x j z i = x i + ( q i /a ii ) x i = ωz i + (1 ω ) x i END FOR r = b A x T c = k r k / k b k k = k + 1 END WHILE
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4. ITERATIVE SOLUTION TECHNIQUES FOR SOLVING
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