5. Accuracy of a difference approximation can be improved by a linear combination of two difference...

   

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5. Accuracy of a difference approximation can be improved by a linear combination of two difference approximations so that the lowest-order (or leading) truncation errors of the two approximations are cancelled. The improved difference approximation is given by: Siz α-Si+1 -2 S₁ + Si-¹ + (1 - α). f" h² fi+2 -2f₁ + fi-2. 4h² (a) Determine the value of a for the following approximation so that the accuracy is optimized or maximized. Also Determine the truncation error. (15 pt) (b) Follow (a) to determine the value f" for f(x)=x² ex at x = 1.0 with h = 0.2 and h = 0.1. In addition, please justify the order of the truncation error. Also compare your results with those obtained in Problem 4. (15 pt) 5. Accuracy of a difference approximation can be improved by a linear combination of two difference approximations so that the lowest-order (or leading) truncation errors of the two approximations are cancelled. The improved difference approximation is given by: Siz α-Si+1 -2 S₁ + Si-¹ + (1 - α). f" h² fi+2 -2f₁ + fi-2. 4h² (a) Determine the value of a for the following approximation so that the accuracy is optimized or maximized. Also Determine the truncation error. (15 pt) (b) Follow (a) to determine the value f" for f(x)=x² ex at x = 1.0 with h = 0.2 and h = 0.1. In addition, please justify the order of the truncation error. Also compare your results with those obtained in Problem 4. (15 pt)