pb2_10 - MPO 662 Problem Set 2 Feel free to use symbolic...

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MPO 662 – Problem Set 2 Feel free to use symbolic computation software such as mathematica or the symbolic toolbox in matlab to carry out the algebra. See 1 to see an example with matlab. 1. Use Taylor series to derive explicit finite difference approximations according to the following specifications: A 3 rd order approximation to the first derivatives using the points i - 2 , i - 1 , i and i + 1. Use the same points as above and derive the highest possible approximation to the third derivative. Derive a 4 th order centered approximations to the second derivative, using points i , i ± 1, and i ± 2. 2. Compact differencing for second derivatives using the formula α u xx | i - 1 + u xx | i + α u xx | i +1 = a 1 u i +1 - 2 u i + u i - 1 Δ x 2 + a 2 u i +2 - 2 u i + u i - 2 x 2 (1) Determine the equations that would make the scheme fourth-order. write the solu- tion in terms of the parameter α . What choice of parameters would minimize the computational stencil on the right hand side?
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